Navigation system applications of sigma-point Kalman filters for nonlinear estimation and sensor fusion

ABSTRACT

A method of estimating the navigational state of a system entails acquiring observation data produced by noisy measurement sensors and providing a probabilistic inference system to combine the observation data with prediction values of the system state space model to estimate the navigational state of the system. The probabilistic inference system is implemented to include a realization of a Gaussian approximate random variable propagation technique performing deterministic sampling without analytic derivative calculations. This technique achieves for the navigational state of the system an estimation accuracy that is greater than that achievable with an extended Kalman filter-based probabilistic inference system.

RELATED APPLICATION

This application claims benefit of U.S. patent application No.60/559,708, filed Apr. 5, 2004.

FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

This invention was made with United States government support pursuantto Grant No. N00014-02-C-0248 from the Office of Naval Research, andGrant No. F33615-98-C-3516 from the Defense Advance Research ProjectsAgency. The United States has certain rights to this invention.

COPYRIGHT NOTICE

© 2005 Oregon Health and Sciences University. A portion of thedisclosure of this patent document contains material that is subject tocopyright protection. The copyright owner has no objection to thefacsimile reproduction by anyone of the patent document or the patentdisclosure, as it appears in the Patent and Trademark Office patent fileor records, but otherwise reserves all copyright rights whatsoever. 37CFR § 1.71(d).

TECHNICAL FIELD

This application relates to estimating the navigational state of asystem and, in particular, to applying a sigma-point Kalman filter(SPKF) based probabilistic inference framework to effect nonlinearestimation and sensor fusion in loosely coupled or tightly coupledintegrated navigation systems.

BACKGROUND OF THE INVENTION

A core objective in the design of integrated navigation systems isoptimal fusing of noisy observations from global positioning systems(GPS), inertial measurement units (IMU), and other sensors. Currentnavigation systems use the extended Kalman filter (EKF) as the standardtechnique for integrating sensor data. In an inertial navigation system(INS), the EKF combines gyroscope and accelerometer data from an IMUwith a kinematics or dynamic model of a vehicle. Other sensors such as abarometric altimeter or magnetic compass may also be integrated.Supplied GPS position and velocity measurements are then integrated withthe INS using the same EKF. Position, attitude, velocities, and sensorbiases are all estimated. In addition, the GPS receiver may employ itsown EKF to solve position and velocity estimates and timing fromsatellite pseudo-range, phase, and Doppler data. Alternatively, in atightly coupled approach, a single EKF may be used to combine rawsatellite signals with the IMU and other sensor measurements to make anoptimal estimation. Unfortunately, the EKF is based on a suboptimalimplementation of the recursive Bayesian estimation framework applied toGaussian random variables. The EKF achieves only first order accuracy inthe calculation of the posterior mean and covariance of the transformedrandom variables. This can seriously affect the accuracy or even lead todivergence of the system.

The basic framework for the EKF entails estimation of the state of adiscrete-time nonlinear dynamic systemx _(k+1) =F(x _(k) ,v _(k))y _(k) =H(x _(k) ,n _(k)),where x_(k) represents the unobserved state of the system and y_(k) isthe only observed signal. The process noise v_(k) drives the dynamicsystem, and the observation noise is given by n_(k). Additivity of thenoise sources is not assumed. The system dynamic model F and H areassumed known. In state estimation, the EKF is the standard method ofchoice to achieve a recursive approximate maximum-likelihood estimationof the state x_(k). Given the noisy observation y_(k), the recursiveestimation for x_(k) can be expressed in the form{circumflex over (x)} _(k)=(optimal prediction of x _(k))+κ_(k) [y_(k)−(optimal prediction of y _(k))].This recursion provides the optimal minimum mean-squared error estimatefor x_(k), assuming the prior estimate of {circumflex over (x)}_(k−1),and current observation y_(k) are Gaussian random variables (GRVs).Linearity of the model need not be assumed. The optimal terms in thisrecursion are given by{circumflex over (x)} _(k) ⁻ =E[F({circumflex over (x)} _(k−1) ,v_(k−1))] κ_(k) =P _(x) _(k) _(y) _(k) P_({tilde over (y)}k{tilde over (y)}k) ⁻¹ ŷ _(k) ⁻ =E[H( {circumflex over(x)} _(k) ⁻ ,n _(k))],where the optimal prediction of x_(k) is written as {circumflex over(x)}_(k) ⁻ and corresponds to the expectation of a nonlinear function ofthe random variables {circumflex over (x)}_(k−1) ⁻ and v_(k−1) (similarinterpretation for the optimal prediction ŷ_(k) ⁻). The optimal gainterm κ_(k) is expressed as a function of posterior covariance matrices(with {tilde over (y)}_(k)=y_(k)−ŷ_(k)). These terms also require takingexpectations of a nonlinear function of the prior state estimates. TheKalman filter calculates these quantities exactly in the linear case andcan be viewed as an efficient method for analytically propagating GRVthrough linear system dynamics. For nonlinear models, however, the EKFapproximates the optimal terms as:{circumflex over (x)} _(k) ⁻ ≈F({circumflex over (x)} _(k−1) , v ) κ_(k)≈{circumflex over (P)} _(x) _(k) _(y) _(k) {circumflex over (P)}_({tilde over (y)}k{tilde over (y)}y) ⁻¹ ŷ _(k) ⁻ ≈H({circumflex over(x)} _(k) ⁻ , n ),where predictions are approximated as simply the function of the priormean value for estimates (no expectation taken). The noise means aredenoted by n=E[n] and v=E[v] and are usually assumed to be equal tozero. The covariances are determined by linearizing the dynamicequations (x_(k+1)≈Ax_(k)+Bv_(k), y_(k)≈Cx_(k)+Dn_(k)) and thendetermining the posterior covariance matrices analytically for thelinear system. In other words, in the EKF, the state distribution isapproximated by a GRV, which is then propagated analytically through the“first-order” linearization of the nonlinear system. As such, the EKFcan be viewed as providing “first-order” approximations to the optimalterms. (While “second-order” versions of the EKF exist, their increasedimplementation and computational complexity tend to prohibit their use.)These approximations can, however, introduce large errors in the trueposterior mean and covariance of the transformed (Gaussian) randomvariable, which may lead to sub-optimal performance and sometimesdivergence of the filter.

Julier, S., Uhlmann, J., and Durrant-Whyte, H., “A New Approach forFiltering Nonlinear Systems,” in Proceedings of the American ControlConference (1995), derived a novel, more accurate, and theoreticallybetter grounded alternative to the EKF called the unscented Kalmanfilter (UKF) for state estimation within the application domain ofnonlinear control. The UKF addresses the approximation issues of theEKF. The state distribution is again represented by a GRV but is nowspecified using a minimal set of carefully chosen sample points, calledsigma points. These sigma points completely capture the true mean andcovariance of the GRV, and when propagated through the true nonlinearsystem, capture the posterior mean and covariance accurately to thethird order (Taylor series expansion) for any nonlinearity. The UKF canbe explained by considering propagating a random variable x (dimensionL) through a nonlinear function, y=g(x), and assuming x has mean x andcovariance P_(x). To calculate the statistics of y, a matrix χ of 2L+1sigma vectors χ_(i) (with corresponding weights W_(i)) is formed,according to the following:χ₀ = xW ₀ ^((m)=λ() L+λ)⁻¹χ_(i) = x +(√{square root over ((L+λ)P _(x))})_(i) i=1, . . . , L W ₀^((c)=λ() L+λ)⁻¹+(1−α²+β)χ_(i) = x−(√{square root over ((L+λ)P _(x))})_(i−L) i=L+1, . . . , 2L W_(i) ^((m)) =W _(i) ^((c))=0.5(L+λ)⁻¹ i=1, . . . , 2Lwhere λ=α²(L+κ)−L is a scaling parameter. α determines the spread of thesigma points around x and is usually set to a small positive value (e.g.1≦α≦1e−3). κ is a secondary, scaling parameter that is usually set to 0,and β is used to incorporate prior knowledge of the distribution of x(for Gaussian distributions, β=2 is optimal). The optimal value of β isa function of the kurtosis of the prior random variable. (√{square rootover ((L+λ)P_(x))})_(i) is the ith row of the matrix square root. Thesesigma vectors are propagated through the nonlinear function

$\begin{matrix}{\chi_{0} = \overset{\_}{x}} & {W_{0}^{(m)} = {\lambda\left( {L + \lambda} \right)}^{- 1}} \\{\chi_{i} = {\overset{\_}{x} + \left( \sqrt{\left( {L + \lambda} \right)P_{x}} \right)_{i}}} & {W_{0}^{(c)} = {{\lambda\left( {L + \lambda} \right)}^{- 1} + \left( {1 - \alpha^{2} + \beta} \right)}} \\{{i = 1},\ldots\mspace{14mu},L} & \; \\{\chi_{i} = {\overset{\_}{x} - \left( \sqrt{\left( {L + \lambda} \right)P_{x}} \right)_{i - L}}} & {W_{i}^{(m)} = {W_{i}^{(c)} = {0.5\left( {L + \lambda} \right)^{- 1}}}} \\{{i = {L + 1}},\ldots\mspace{14mu},{2L}} & {{i = 1},\ldots\mspace{14mu},{2L}}\end{matrix}$and the mean and covariance of y are approximated using a weightedsample mean and covariance of the posterior sigma points

$\begin{matrix}{\overset{\_}{y} \approx {\sum\limits_{i = 0}^{2L}{W_{i}^{(m)}\Upsilon_{i}}}} & \; & \; & {P_{y} \approx {\sum\limits_{i = 0}^{2L}{{W_{i}^{(c)}\left( {\Upsilon_{i} - \overset{\_}{y}} \right)}{\left( {\Upsilon_{i} - \overset{\_}{y}} \right)^{T}.}}}}\end{matrix}$

This specific implementation of the general sigma-point approach forcalculating posterior statistics, is called the unscented transformation(UT) [Julier 1995]. The sigma-point approach differs substantially fromgeneral “sampling” methods (e.g., Monte-Carlo methods and grid-basedfilters), which require orders of magnitude more sample points in anattempt to propagate an accurate (possibly non-Gaussian) distribution ofthe state. The deceptively simple approach results in approximationsthat are accurate to the third order for Gaussian inputs for allnon-linearities. For non-Gaussian inputs, approximations are accurate toat least the second-order, with the accuracy of third and higher ordermoments determined by the choice of α and β. A simple example is shownin FIG. 1 for a two-dimensional system. The left-hand plots show thetrue mean and covariance propagation using Monte-Carlo sampling; themiddle plots show the results using a linearization approach as would bedone in the EKF; and the right-hand plots show the performance of thesigma-point approach (only 5 sigma points required for L=2). Thesuperior performance of the sigma-point approach is clear.

The sigma-point Kalman filter (SPKF) is a better alternative to the EKFfor Gaussian approximate probabilistic inference in general nonlineardynamic state-space models. SPKF is a family of Kalman filters usingderivativeless deterministic sampling for Gaussian approximate nonlinearestimation. The sigma-point approach is the underlying unifyingtechnique that is common to SPKFs and is a method of calculating thestatistics of a random variable that undergoes a nonlineartransformation. These calculations form the core of the optimal Kalmantime and measurement equations, which are simply the original (optimal)recursive Bayesian estimation integral equations recast under a Gaussianassumption.

The sigma-point approach for approximating the statistics of a randomvariable that undergoes a nonlinear transformation is summarized by thefollowing three steps:

1. A set of weighted sigma-points is deterministically calculated usingthe mean and square-root decomposition of the covariance matrix of theprior random variable. As a minimal requirement, the sigma-point setcompletely captures the first and second order moments of the priorrandom variable. Higher order moments can be captured, if so desired, atthe cost of using more sigma-points.

2. The sigma-points are then propagated through the true nonlinearfunction using functional evaluations alone, i.e., no analyticalderivatives are used, to generate a posterior sigma-point set.

3. The posterior statistics are calculated (approximated) usingtractable functions of the propagated sigma-points and weights.

When the UT implementation of the sigma-point approach is used, theresulting filter is the UKF. The UKF is one specific implementation ofSPKF.

The superior performance of the sigma-point approach would makedesirable an application of SPKF to nonlinear estimation and sensorfusion in the operation of vehicle navigation systems.

SUMMARY OF THE INVENTION

An object of the invention is to provide a method of estimating thenavigational state of a system with an accuracy that is greater thanthat achievable with an extended Kalman filter-based probabilisticinference system.

The present invention is a method of estimating the navigational stateof a system. The navigational state is characterized by random variablesof a state space model that specifies a time evolution of the system andits relationship to sensor observations. The method entails acquiringobservation data produced by measurement sensors that provide noisyinformation about the navigational state of the system and providing aprobabilistic inference system to combine the observation data withprediction values of the system state space model to estimate thenavigational state of the system. The probabilistic inference system isimplemented to include a realization of a Gaussian approximate randomvariable propagation technique performing deterministic sampling withoutanalytic derivative calculations. This technique achieves for thenavigational state of the system an estimation accuracy that is greaterthan that achievable with an extended Kalman filter-based probabilisticinference system.

In a preferred embodiment, the measurement sensors include an inertialmeasurement unit (IMU) that operates to estimate a set of navigationalstate components. The navigational state component set includes angularvelocity and attitude information relating to an object in motion andbias and scale factor information associated with the IMU. The randomvariable propagation technique of the probabilistic inference systemused to estimate the navigational state of the system includes usingmean values and square root decomposition of a covariance matrix of aprior random variable to deterministically calculate a set ofsigma-points and an associated set of weight values. The calculated setof weighted sigma-points is propagated through a nonlinear state spacemodel to produce a posterior set of weighted sigma-points used tocalculate posterior statistics. The calculation of posterior statisticsentails the use of closed form functions of the propagated sigma-pointsand weight values. The calculated posterior statistics give anapproximately optimal update of the navigational state and itscovariance by combining the prediction values of the system state spacemodel and the sensor observations.

A preferred embodiment implements a SPKF based latency delayed sensorfusion technique for situations in which the observation data providenoisy information about the navigational state of the system at a timethat lags a current time because of sensor latency. The method entailsmaintaining a cross correlation between a navigational state of thesystem at a current time and the navigational state to which the latencylagged observation data correspond and uses the cross correlation tooptimally combine latency lagged observation data with the predictionvalues of the system state space model.

Additional aspects and advantages of this invention will be apparentfrom the following detailed description of preferred embodiments, whichproceeds with reference to the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows the accuracy of a scaled unscented transformation for meanand covariance propagation of a Gaussian random variable through anarbitrary nonlinear function.

FIG. 2 is a pictorial block diagram of an unmanned aerial vehicle (UAV)guidance, navigation, and control (GNC) system in which a preferredembodiment of the invention is implemented.

FIG. 3 is a schematic representation of sensor inherent latencymanifested within a Kalman filter based estimation framework.

FIG. 4 is a process flow diagram showing the calculation of thecross-covariance using the sigma-point approach for sensor latencycompensation.

FIG. 5 is a graphical rendering of the helicopter flight trajectory usedfor state estimation experiments.

FIG. 6 is a set of graphical plots showing the error performance of thehelicopter flown in simulation along the trajectory of FIG. 5.

FIG. 7 shows for EKF and SPKF the desired flight trajectory, truerealized trajectory, and estimated flight trajectory of a high speednose-in turn performed by the helicopter flown in simulation.

FIG. 8 shows a comparison of RMSE performance versus increase in IMUadditive noise.

FIG. 9 shows the comparative relationship of the estimatedthree-dimensional positions for SPKF- and EKF-based systems on actualtest flight telemetry.

FIG. 10 shows a two-dimensional (top down) projection of the estimationresults presented in FIG. 9.

FIGS. 11, 11B, and 11C show the effect of GPS outages on, respectively,the separate north, east, and down components of the estimatedthree-dimensional positions presented in FIG. 9.

FIG. 12 is a schematic diagram presented as a graphical aid inexplanation of the Gaussian Mixture Sigma-Point Particle Filter (GMSPPF)algorithm.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

A preferred embodiment of the method of estimating the navigationalstate of a system is carried out in application of the SPKF to a looselycoupled implementation of guidance and navigation system (GNS) forunmanned aerial vehicle autonomy (UAV), specifically an autonomoussmall-scale helicopter. The core component of such a UAV is ahigh-performance digital computer based guidance, navigation, andcontrol (GNC) system as shown schematically in FIG. 2. The mainsubsystems of the GNC system are a control system (CS) and a guidanceand navigation system (GNS). The GNS takes noisy avionics sensormeasurements (e.g., GPS, IMU, and altimeter) as inputs, and then fusesthem in a probabilistic sense with predictions from a vehicle dynamicsmodel to calculate optimal vehicle navigation state solutions. The term“fuse” refers to the Kalman optimal combination of the prior stateestimate with the new information contained in the sensor measurement.This is achieved through the Kalman measurement-update step. These stateestimates, together with desired flight trajectories, are then appliedto the control system, which computes some form of optimal control lawto drive the flight surface actuators of the vehicle. This GNS isconsidered a loosely coupled GPS/INS (inertial navigation system)integration because the processing internal to the GPS is doneindependently of other sensor measurements. A computer readable memorystoring instructions for configuring a signal processing system ispreferably used to perform the method. The computer readable memory caninclude magnetic media (such as floppy disk or a hard drive), opticalmedia (such as a CD or a DVD) or electronic media (such as rewritable ornon-rewritable memory).

The current state-of-the-art probabilistic inference systems used forsuch UAV guidance and navigation systems are usually EKF based. Theexperimental platform, which was developed by the Aerial Robotics Groupat the MIT Laboratory for Information and Decision Systems, was builtaround an EKF based GNS implementation. Real-world experimentalverification of the operation of the SPKF entailed replacement of theEKF in the UAV-GNC system by a SPKF and comparison of its performance toan existing EKF base-line system, with specific focus on: 1) improvedsix-degrees-of-freedom (6DOF) state estimation accuracy, 2) SPKF basedcompensation for GPS latency, 3) evaluation of improved control enveloperesulting from use of a better state estimator, and 4) robustness to IMUdegradation. The vehicle dynamic process model and the avionics sensormodels for use inside the SPKF based state estimator are describedbelow.

Vehicle State-Space Process Model: Because of its high computationalcomplexity, a high-fidelity nonlinear dynamic model of UAV movement isnot ideally suited for use within the navigation filter loop. (TheExperimental Results section presented below describes use of ahigh-fidelity (70 parameters, 43 states) nonlinear dynamic model of UAVmovement for UAV simulators and control system design.) For this reason,applicants opted for the standard IMU driven kinematic process modelformulation that comprises an INS mechanization component and a IMUsensor error model component. Because low cost MEMS-based IMUs such asthe one used in the avionics system have large bias and scale factorerrors, these components are included into the state vector to beestimated.

The estimated values of these error components are then used to correctthe raw IMU acceleration and gyroscope-rate measurements before they areused inside the INS mechanization equations of the process model. The 16dimensional state vector of this model is given byx=[p^(T)v^(T)e^(T)a_(b) ^(T)ω_(b) ^(T)],where p=[x y z]^(T) and v=[v_(x) v_(y) v_(z)]^(T) are the respectiveposition and velocity vectors of the vehicle in the navigation frame,e=[e₀ e₁ e₂ e₃]^(T) is the unity norm vehicle attitude quaternion,a_(b)=[a_(x) _(b) a_(y) _(b) a_(z) _(b) ]^(T) is the vector of IMUacceleration biases, and ω_(b)=[p_(b) q_(b) r_(b)]^(T) is the IMUgyroscope rate bias vector. Although a separate scale factor can beincluded in addition to the bias term in the state vector, applicants'experiments demonstrated that it is sufficient to model the combinedeffect of the bias and scale error terms as a single time-varying biasterm.

The continuous time kinematic navigation equations (INS mechanizationequations and error model) operating on this state vector and driven bythe error corrected IMU measurements are given below:

$\begin{matrix}{\overset{.}{p} = v} & (1) \\{\overset{.}{v} = {{C_{b}^{n}\left( {\overset{\_}{a} - a_{{\overset{\sim}{r}}_{imu}}} \right)} + {\begin{bmatrix}0 & 0 & 1\end{bmatrix}^{T}\mspace{11mu} g}}} & (2) \\{\overset{.}{e} = {{- \frac{1}{2}}{\overset{\sim}{\Omega}}_{\overset{\_}{\omega}}e}} & (3) \\{{\overset{.}{a}}_{b} = w_{a_{b_{k}}}} & (4) \\{{\overset{.}{\omega}}_{b} = {w_{\omega_{b_{k}}}.}} & (5)\end{matrix}$C_(b) ^(n) is the direction cosine matrix (DCM) transforming vectorsfrom the body frame to the navigation frame. The DCM is a nonlinearfunction of the current attitude quaternion and is given by

$C_{b}^{n} = {\left( C_{n}^{b} \right)^{T} = {{2\;\begin{bmatrix}{0.5 - e_{2}^{2} - e_{3}^{2}} & {{e_{1}e_{2}} - {e_{0}e_{3}}} & {{e_{1}e_{3}} + {e_{0}e_{2}}} \\{{e_{1}e_{2}} + {e_{0}e_{3}}} & {0.5 - e_{1}^{2} - e_{3}^{2}} & {{e_{2}e_{3}} - {e_{0}e_{1}}} \\{{e_{1}e_{3}} - {e_{0}e_{2}}} & {{e_{2}e_{3}} + {e_{0}e_{1}}} & {0.5 - e_{1}^{2} - e_{2}^{2}}\end{bmatrix}}.}}$The term g is the gravitational acceleration component, and ā and ω arethe bias and noise corrected IMU accelerometer and gyroscope ratemeasurements, i.e.,ā=ã−a _(b) −n _(a)ω={tilde over (ω)}−ω_(b) −C _(n) ^(b)ω_(c) −n _(ω).

In the above equations, ā and ω are the raw measurements coming from theIMU, n_(a) and n_(ω) are the IMU acceleration and gyroscope-ratemeasurement noise terms, and ω_(c) is the rotational rate of the earthas measured in the navigation frame (Coriolis effect). In general, ω_(c)is a function of the location of the navigational frame relative to theearth frame and hence is time-varying as the navigation frame movesrelative to the earth frame. However, for their purposes (aggressiveautonomous UAV flight within a very small airspace volume), applicantsassumed that the navigation frame does not change relative to the earthframe, resulting in a constant ω_(c) for a given origin location(lat/long) of the navigation frame. {tilde over (Ω)}_({tilde over (ω)})is a 4×4 skew-symmetric matrix composed of the error corrected IMUgyro-rate measurements, i.e.,

${\overset{\sim}{\Omega}}_{\overset{\_}{\omega}} = {\begin{bmatrix}0 & {\;{\overset{\_}{\omega}}_{p}} & {\;{\overset{\_}{\omega}}_{q}} & {\;{\overset{\_}{\omega}}_{r}} \\{- {\overset{\_}{\omega}}_{p}} & 0 & {- {\overset{\_}{\omega}}_{r}} & {\;{\overset{\_}{\omega}}_{q}} \\{- {\overset{\_}{\omega}}_{q}} & {\;{\overset{\_}{\omega}}_{r}} & 0 & {- {\overset{\_}{\omega}}_{p}} \\{- {\overset{\_}{\omega}}_{r}} & {- {\overset{\_}{\omega}}_{q}} & {\;{\overset{\_}{\omega}}_{p}} & 0\end{bmatrix}.}$In Eq. (2), a_({tilde over (r)}) _(imu) is the IMU-lever-arm couplingcomponent resulting from locating the IMU not at the center of gravityof the vehicle. This component can be ignored if the navigation filtercomputes the state estimate at the IMU location. This IMU centricnavigation solution can then simply be transformed to the center ofgravity location after the fact as needed by the vehicle control system.

The final components of the process model, Eqs. (4) and (5), model thetime-varying nature of the IMU sensor bias error terms. Usually, sensorerrors in an INS are modeled as a zero-mean, stationary, first-orderGauss-Markov process. Since the biases and scale factors of low costMEMS-based IMU sensors exhibit non-zero mean and non-stationarybehavior, the errors are modeled as a random walk to improve thetracking of these time-varying errors by the navigation filter. Thisdoes require, however, that the effect of these errors be observablethrough the specific choice of measurement model.

The position and velocity discrete-time updates are calculated by thefollowing simple first-order Euler updatep _(k+1) =p _(k) +{dot over (p)} _(k) ·dtv _(k+1) =v _(k) +{dot over (v)} _(k) ·dt,where {dot over (p)}_(k) and {dot over (v)}_(k) are calculated usingEqs. (1) and (2) and dt is the integration time-step of the system (inthis system the integration time-step was dictated by the IMU rate,i.e., dt=10 ms). The quaternion propagation equation can be expressed indiscrete form with an analytical calculation of the exponent of theskew-symmetric matrix given by Stevens, B. and Lewis, F., AircraftControl and Simulation, New York, N.Y.: Wiley, 1992. The discrete-timeupdate can be written as

$\begin{matrix}{e_{k + 1} = {{\exp\left( {{- \frac{1}{2}}{\overset{\sim}{\Omega} \cdot d}\; t} \right)}\mspace{11mu}{e_{k}.}}} & (6)\end{matrix}$Further denotingΔφ= ω _(p) ·dtΔθ= ω _(q) ·dtΔψ= ω _(r) ·dt,as effective rotations around the (body frame) roll, pitch, and yaw axesundergone by the vehicle during the time period dt, assuming that theangular rates ω _(p), ω _(q), and ω _(r) remained constant during thatinterval, one can introduce the 4×4 skew-symmetric matrix

$\Phi_{\Delta} = {{{\overset{\sim}{\Omega} \cdot d}\; t} = {\begin{bmatrix}0 & {\Delta\;\phi} & {\Delta\;\theta} & {\Delta\;\psi} \\{{- \Delta}\;\phi} & 0 & {{- \Delta}\;\psi} & {\Delta\;\theta} \\{{- \Delta}\;\theta} & {\Delta\;\psi} & 0 & {{- \Delta}\;\phi} \\{{- \Delta}\;\psi} & {{- \Delta}\;\theta} & {\Delta\;\phi} & 0\end{bmatrix}.}}$Using the definition of the matrix exponent and the skew symmetricproperty of Φ_(Δ), one can write down the following closed-formsolution:

$\begin{matrix}\begin{matrix}{{{\exp\left( {{- \frac{1}{2}}\Phi_{\Delta}} \right)} = {{I\mspace{11mu}{\cos(s)}} - {\frac{1}{2}\Phi_{\Delta}\;\frac{\sin(s)}{s}}}},} \\{where} \\{s = {{\frac{1}{2}{\left\lbrack {\Delta\;\phi\mspace{14mu}\Delta\;\theta\mspace{14mu}\Delta\;\psi} \right\rbrack }} = {\frac{1}{2}{\sqrt{\left( {\Delta\;\phi} \right)^{2} + \left( {\Delta\;\theta} \right)^{2} + \left( {\Delta\;\psi} \right)^{2}}.}}}}\end{matrix} & (7)\end{matrix}$Eqs. (6) and (7) ensure (at least theoretically) that the updatedquaternion e_(k+1) has a unit norm. However, a small Lagrange multiplierterm can be added to the first component of Equation 7 to furthermaintain numerical stability and the unity norm of the resultingquaternion. The resulting final solution for the time-update of thequaternion vector is given by

${e_{k + 1} = \left\lbrack {{I\;\left( {{\cos(s)} + {{\eta \cdot d}\;{t \cdot \lambda}}} \right)} - {\frac{1}{2}\Phi_{\Delta}\;\frac{\sin(s)}{s}e_{k}}} \right\rbrack},$where

λ = 1 − e_(k)²is the deviation of the square of the quaternion norm from unityresulting from numerical integration errors, and η is the factor thatdetermines the convergence speed of the numerical error. These factorsserve the role of the above mentioned Lagrange multiplier that ensuresthat the norm of the quaternion remains close to unity. The constrainton the speed of convergence for stability of the numerical solution isη·dt<1.

Finally, the discrete time random-walk process for the IMU sensor errorterms is given bya _(b) _(k+1) =a _(b) _(k) +dt·w _(a) _(bk)ω_(b) _(k+1) =ω_(b) _(k) +dt·w _(ω) _(bk) ,where w_(a) _(bk) and w_(ω) _(bk) are zero-mean Gaussian randomvariables.

These navigation equations are considered a direct formulation, asopposed to the alternative indirect (error) formulation. This choice wasmade for consistency with the MIT EKF implementation mentioned above.

Observation Models: The system made use of two independent avionicsensors to aid the INS: a 10 Hz, 50 ms latency GPS (Ashtech G12) and abarometric altimeter that measures absolute altitude as a function ofambient air pressure. The observation models used in our system forthese sensors (described below) are highly nonlinear, making the use ofthe SPKF framework again preferable to an EKF solution.

GPS: Since our GPS antenna is not located at the same location in thebody frame as the IMU, it not only observes the position and velocity ofthe body in the navigation frame, but also the attitude of the bodyrelative to the navigation frame because of the “lever-arm effect.” Morespecifically, the GPS observation model is given by:p _(k) ^(GPS) =p _(k−N) +C _(b) ^(n) {tilde over (r)} _(gps) +n_(pk)  (8)v _(k) ^(GPS) =v _(k−N) +C _(b) ^(n)ω_(k−N) ×{tilde over (r)} _(gps) +n_(v) _(k)   (9)where p_(k−N) and v_(k−N) are the time-delayed (by N samples due tosensor latency) 3D navigation-frame position and velocity vectors of thevehicle, {tilde over (r)}_(gps) is the location of the GPS antenna inthe body frame (relative to the IMU location), ω_(k−N) are the truerotational rates of the vehicle at time k−N, and n_(pk) and n_(v) _(k)are stochastic measurement noise terms. The noise terms are modeled asbeing time-dependent because the accuracy of observations varies overtime according to the current PDOP value of the loosely coupled GPSsolution. Since the DCM, C_(b) ^(n), in Eqs. (8) and (9) are a functionof the attitude quaternion, the GPS measurements provide information notonly of the position and velocity of the vehicle, but also of itsattitude. This removes the need for an absolute attitude sensor such asa magnetic compass or tilt-sensor. However, this will also result in thenon-observability of the IMU sensor errors during prolonged periods ofGPS outages, which in turn can lead to significant INS drift.

The time delay (N samples) in the GPS model equations results from theinternal GPS processing latency inherent to all loosely coupled GPSsolutions. This implies that the latest GPS measurement relates to thestate of the vehicle as it was a number of samples in the past. If thespecific latency of the GPS is small, it can (and often is) ignored. Ifthe latency is significant, however, care is taken when fusing thislagged information with the current estimate of the state of the vehiclein the measurement update step of the Kalman filter.

Barometric altimeter: Ambient air pressure provides an accurate sourceof sea-level altitude information. Important sources of error are sensorquantization and measurement noise. A high-end altimeter with 10⁻³ psi(0.6 meter) resolution was used. The measurement noise was assumed to bezero-mean, white, and Gaussian. The observation model that incorporatesthese effects are:

${z_{k}^{alt} = {{- \frac{1}{\varphi}}\;{\ln\left\lbrack \frac{\rho_{0}^{q}\left\lfloor {\left( {{\rho_{0}\;{\exp\left( {\varphi \cdot z_{k}} \right)}} + n_{z_{a}}} \right)/\rho_{0}^{q}} \right\rfloor}{\rho_{0}} \right\rbrack}}},$where ρ₀ is the nominal air pressure at sea-level, φ is the pressuredecay rate with altitude constant (1.16603×10⁻⁴ psi/m), z_(k) is thecurrent navigation-frame z-axis position of the vehicle, ρ₀ ^(q) is theair pressure quantization resolution of the altimeter (10⁻³ psi), z_(k)^(alt) is the altimeter output and └•┘ is the integer flooring function.This model is not only a nonlinear function of the state, but themeasurement noise also effects the output altitude measurement in anon-additive fashion. Again, for such a model the use of the SPKF notonly allows for a much simpler implementation than the EKF (noanalytical derivatives need to be calculated), but also results in moreaccurate estimation results. Nonlinear sensor effects such asquantization and compressions are all readily incorporated within theKalman estimation framework using the inherently nonlinear modelingcapability of SPKFs.

The MIT designed, fine-tuned EKF was replaced in the helicopter vehiclewith a SPKF based state estimator to implement the present invention.The state estimator uses the above-described nonlinear kinematic processmodel and observation models of the GPS and barometric altimeter. Thestate estimator is of a square-root central difference Kalman filter(SR-CDKF) SPKF formulation because of its ease of implementation,intuitive choice of scaling parameters, and numerical robustness. TheSR-CDKF algorithm is set forth below as Algorithm 4.

There are a number of problem specific issues that are dealt with inadapting the general SPKF framework to the UAV state estimation problem.The important ones include asynchronicity, differing sample rates, andtime-varying dimension of sensor observations; filter initialization;GPS latency; and quaternion unity norm constraint.

The first issue is measurement asynchronicity, varying sample rates, andtime-varying dimensions. Unlike well behaved synthetic laboratoryexperiments, the UAV avionics system operates on an asynchronous messagepassing principle. Although the main system (and filter) is clocked atthe IMU rate (100 Hz), the avionic sensors (GPS and altimeter) operatenot only asynchronously from this clock, but their measurements are alsoprovided at different sampling rates (10 Hz and 20 Hz respectively).This in turn implies that every filter cycle does not necessarily haveboth a time-update and a measurement-update step. Every time a new IMUmeasurement becomes available (roughly every 10 ms), the filterundergoes a time-update using the IMU driven kinematic process model,but a measurement-update is done only if there are actually new sensordata available for processing. The flight-computer polls thenavigational sensors (or their simulation equivalents) to determinewhether new data are available, and if so, update a bank of sensorbuffers with these new measurements. These measurements are accompaniedby a unique time-stamp if new data were written into the sensor buffers.

Based on these time-stamps and prior knowledge of the different updaterates of the sensors, the estimator system builds up an adaptiveevent-map of when to expect new data from the different sensors. Thisevent-map, which is built up during the filter initialization stage, isused in dealing with the GPS latency problem.

Since the different sensors have different observation vector dimensions(GPS has six dimensions, and altimeter has one dimension) and operate atdifferent rates, the SPKF observation model for any given measurementupdate is configured to adapt to this time-varying total observationdimension. If both sensors report new data, the effective observationmodel (and its measurement noise random variable) will be aconcatenation of the individual sensor observation models andsigma-points will be drawn from this augmented observation state (true16D state and augmented noise state). It might also be the case thatonly one sensor (either the GPS or altimeter) reports new data for agiven measurement update, resulting in the observation model (and therelated SPKF based measurement update sigma-point generation) revertingto the relevant single sensor model. This time-varying observation modelrequires careful book-keeping of sensor events to accurately adapt thefilter to the time-varying nature of the sensor data stream.

The second issue is filter initialization. During the initialization ofthe navigation filter (state estimator), sensor data are observed andprocessed over a number of seconds while the UAV is at rest on theground in a known position and attitude. This step contributes toensuring robust performance of the filter during subsequent aggressivemaneuvers.

The initial position estimate, {circumflex over (P)}₀=[{circumflex over(x)}₀ŷ₀{circumflex over (z)}₀]^(T), in the NED frame is based solely onthe averaged GPS position measurements for the first couple of secondswhile the UAV is stationary. The initial NED frame velocity estimate,{circumflex over (v)}₀=[{circumflex over (v)}_(x) ₀ {circumflex over(v)}_(y) ₀ {circumflex over (v)}_(z) ₀ ]₀, is initialized to zero. Thehelicopter is oriented into a known yaw direction (either due north ordue south) with the x−y (horizontal) plane level (using two spiritlevels) during the initialization phase. This allows the initialquaternion estimate, ê₀=[ê₀ ₀ ê₁ ₀ ê₂ ₀ ê₃ ₀ ]^(T), to be set to a knownvalue such as ê₀=[1 0 0 0]^(T). The averaged IMU accelerometer andgyroscope readings during the initialization period are used toinitialize the estimates of the IMU bias variables. If the IMU isunbiased, the expected stationary accelerometer reading should be a=[00−g]^(T) (where g is the local gravitational acceleration constant,i.e., 9.81 m/s²) and the gyroscope rate readings (ignoring the rotationof the earth) should be zero. Any reading deviating from these expectedvalues can then be used to initialize the bias estimates.

The final important house-keeping task performed during theinitialization phase is the build up the “event map” for when to expectnew data from the avionics sensors. This is done by monitoring thetime-stamps of the sensor stream buffers over a period of time while thehelicopter is stationary. A pre-specified number (≈10) of sensor cyclesare observed before the full event-map is robustly initialized. This inturn implies that the duration of this specific initialization phase isdetermined by the slowest sensor in the avionics system (in this casethe GPS). As stated earlier, an accurate event map is also needed toaccurately compensate for the inherent processing latency present in theGPS measurements.

The third issue is GPS latency. One of the big challenges in building arobust state estimator is dealing with the inherent measurement latencyof the GPS sensor. As previously mentioned, a GPS sensor has a finiteprocessing delay between when a GPS satellite signal is received forprocessing and when the actual position and velocity measurement relatedto that signal becomes available. This implies that the current GPSreading actually corresponds to the position and velocity state of thevehicle at some point in the past. This time difference is called themeasurement latency. For cheaper lower performance GPS systems, thislatency can be in the order of couple of seconds, causing seriousproblems when these measurements are fused inside a Kalman filter withthe current prediction of the vehicle state. FIG. 3 demonstrates thisissue graphically for a linear dynamic state-space model given byx _(k+1) =A _(k) x _(k) +B _(k) u _(k) +v _(k)  (10)y _(k) =C _(k) x _(k) +n _(k,)  (11)where v_(k)˜N(O,R_(v)) and n_(k)˜N(O,R_(n)). The state estimationfilter, in general, receives measurements from a variety of sensors ateach measurement-update step. Some measurements corresponds to thesystem state at the current time, y_(k), given by Equation (2), whileother latency-delayed measurements, y_(k)*, correspond to the systemstate at time l=k−N, i.e., y_(k)*=C_(l)*x_(l)+n_(k)*, where N is thesensor latency measured in sample periods, C_(l)* is the measurementsensitivity matrix, and n_(k)* is the observation noise for the delayedmeasurement with n_(k)˜N(O,R_(n)).

FIG. 3 shows, for a system with delayed measurement resulting fromsensor latency, that at time k, the state estimation filter receives twosets of sensor measurements from the system. A normal measurement y_(k)corresponds to the system state at time k, and a delayed measurementy_(k)* corresponds to the system state at time l=k−N. The followingpresents a technique for optimally fusing these different sets ofmeasurements with the current estimate of the system state.

SPKF based Sensor Latency Compensation: To fuse new information(innovation signal) resulting from a lagged sensor measurement, {tildeover (y)}_(k−n), with the current prediction of the system state,{circumflex over (x)}_(k) ⁻, the following form of the Kalmanmeasurement update equations is used:{circumflex over (x)} _(k) ={circumflex over (x)} _(k) ⁻+κ_(k){tildeover (y)}_(k−n),κ_(k) =P _(x) _(k)_({tilde over (y)}k−n)(P_({tilde over (y)}k−n))⁻¹ ,n=N _(lat).A key insight here is that the optimal Kalman gain in this case is afunction of the cross-covariance between the current state and thelatency lagged system state at the time the sensor measurement was made,P_(x) _(k) _({tilde over (y)}k−n). This quantity can be calculated usingthe sigma-point approach if the following state estimate andcross-covariance terms can be maintained accurately during the latencyperiod:{circumflex over (x)} _(k−n) ,P _(x) _(k) _(x) _(k−n) ⁻ =E[(x _(k)−{circumflex over (x)} _(k) ⁻)(x_(k−n) −{circumflex over (x)}_(k−n))^(T)].The needed term in the optimal Kalman gain expression is then implicitlycalculated using the sigma-point approach as shown in FIG. 4. Tomaintain the correct lagged state and lagged cross-covariance terms, oneredefines the state-space model (process and observation equations) usedinternally by the SPKF. This is done by first augmenting the systemstate with the lagged state that must be maintained, i.e., x_(k)^(a)=[x_(k) ^(T)x_(l) ^(T)]^(T). This is done when the lagged sensormakes its measurement, N_(lat) discrete time periods before themeasurement is available at the output of the sensor. The redefinedsystem state-space model is given by:

$\begin{matrix}{x_{k + 1}^{a} = \begin{bmatrix}{f\left( x_{k} \right)} \\x_{l}\end{bmatrix}} & \; & {y_{k} = {{\overset{\Cup}{h}\left( x_{k}^{a} \right)} = \left\{ {\begin{matrix}{h_{1}\left( x_{k} \right)} & {k \neq {l + N_{lat}}} \\{h_{2}\left( x_{l} \right)} & {k \neq {l + N_{lat}}}\end{matrix}.} \right.}}\end{matrix}$

The top half of the augmented state evolves according to the originalnonlinear process model, while the bottom half (system state atmeasurement instance) is maintained at its initialized value during thelatency period. The observation model states that normal (non latencylagged) sensor observations are functions of the normal propagatedsystem state, i.e., h₁(●), whereas the latency lagged sensor (e.g.,GPS), is a function of the bottom part of the augmented state, thelagged (maintained) value, i.e., it uses h₂(●). By redefining the systemmodels as described above, and applying the normal SPKF filter time- andmeasurement-update equations throughout the latency period, thepredicted augmented system state and covariance will have the followingstructure when the lagged sensor finally “reports in” at the end of thelatency period at l=k−N_(lat):

$\begin{matrix}{{\hat{x}}_{k}^{a -} = \begin{bmatrix}{\hat{x}}_{k}^{-} \\{\hat{x}}_{l}^{*}\end{bmatrix}} & \; & {P_{x_{k}^{a}}^{-} = \begin{bmatrix}P_{x_{k}x_{k}}^{-} & P_{x_{k}x_{l}}^{-} \\P_{x_{l}x_{k}}^{-} & P_{x_{l}x_{l}}^{-}\end{bmatrix}}\end{matrix}$Performing a measurement-update at this point based on the laggedinnovation signal will result in the following gain term:

$\begin{matrix}{{P_{x_{k}^{a}{\overset{\sim}{y}}_{l}} = \begin{bmatrix}P_{x_{k}{\overset{\sim}{y}}_{l}} \\P_{x_{l}{\overset{\sim}{y}}_{l}}\end{bmatrix}},} & {{\kappa_{k} = {{P_{x_{k}^{a}{\overset{\sim}{y}}_{l}}\left( P_{{\overset{\sim}{y}}_{l}} \right)}^{- 1} = \begin{bmatrix}P_{x_{k}{\overset{\sim}{y}}_{l}}^{-} & P_{{\overset{\sim}{y}}_{l}}^{- 1} \\P_{x_{l}{\overset{\sim}{y}}_{l}}^{-} & P_{{\overset{\sim}{y}}_{l}}^{- 1}\end{bmatrix}}},}\end{matrix}$which contains the correct cross-covariance components needed to performthe fusion of the lagged innovation signal with the current systemstate. After this step, the bottom part of the augmented system state isre-initialized with the correctly updated state estimate, inanticipation of the next lagged-sensor cycle.

The fourth issue is SPKF based quaternion unity norm constraint. Asmentioned earlier, the quaternion based attitude formulation has manybenefits over an Euler-angle based formulation. These benefits do,however, come with the caveat that the unity norm of the estimatedquaternion vector has to be maintained during the estimation process.There is an elegant method available to enforce this unity constraintthat is ideally suited to the SPKF framework.

The pseudo-observation of the quaternion sub-vector of the full statevector is defined as

$\begin{matrix}\begin{matrix}{z_{e} = {e}^{2}} \\{= {e^{T}e}} \\{{= {e_{0}^{2} + e_{1}^{2} + e_{2}^{2} + e_{3}^{2}}},}\end{matrix} & (12)\end{matrix}$i.e., the squared norm of the quaternion vector is directly observed.This is a pseudo-observation, however, because there is no actual sensorthat can measure the norm of the true attitude quaternion. The trueattitude quaternion should, however, always have a unity norm so thatthe observation can be forced at time k to be equal to one, i.e.,z_(e,k)≡1One can now calculate the difference (innovation signal) between theassumed known (observed) quaternion norm and that of the prediction ofthe observation based on the current estimate of the quaternion, i.e.,

$\begin{matrix}\begin{matrix}{{\overset{\sim}{z}}_{e,k} = {z_{e,k} - {\hat{z}}_{e,k}^{-}}} \\{= {1 - {E\left\lbrack {{\hat{e}}_{k}^{-}}^{2} \right\rbrack}}} \\{{= {1 - {E\left\lbrack {{\hat{e}}_{0,k}^{2} + {\hat{e}}_{1,k}^{2} + {\hat{e}}_{2,k}^{2} + {\hat{e}}_{3,k}^{2}} \right\rbrack}}},}\end{matrix} & (13)\end{matrix}$and update the vehicle state using a standard Kalman frameworkmeasurement update, i.e.,{circumflex over (x)} _(k) ={circumflex over (x)} _(k) ⁻ +K _(e) z_(e,k) −{circumflex over (z)} _(e,k) ⁻.This step can readily be incorporated into the the existing SPKFframework by simply augmenting the observation model with Equation (12)and concatenating the assumed unity observation to the end of the trueobservation vector. It is also convenient at this point to introduce asynthetic additive observation noise term, n_(e), into thepseudo-observation equation for the quaternion norm,z _(e) =∥e∥ ² +n _(e),where n_(e) is modeled as a zero-mean Gaussian random variable, i.e.,n _(e) ˜N(n _(e);0,σ_(ne) ²),where σ_(n) _(e) ² is the variance. The magnitude of the variance ofthis synthetic noise term will directly affect the “weighting” that thequaternion norm unity constraint receives in relation to the otherobservations. If the variance is set to zero, the SPKF will put a largeweight (importance) on ensuring that the constraint is met, possibly tothe detriment of other components in the state vector estimate. On theother hand, if the variance is set very high, the filter will largelydiscount the pseudo observation, resulting in poor quaternionregulation. The exact setting of this parameter should thus bedetermined in relation to the variances of the other observation noiseterms in order to find an acceptable trade-off between the two extremeconditions.

Because of the quadratic form of the pseudo-observation nonlinearity,the SPKF (in contrast to the EKF) is ideally suited for this specificmethod of quaternion norm regulation. The SPKF generates exact resultsfor quadratic nonlinearities, whereas the EKF results for the samenonlinearity are highly biased and inconsistent. Furthermore, the righthand term of Equation (13), E[ê_(0,k) ²+ê_(1,k) ²+ê_(2,k) ²+ê_(3,k) ²],can be calculated accurately at almost no extra computational cost bynoticing that

$\begin{matrix}{{E\left\lbrack {{\hat{e}}_{0,k}^{2} + {\hat{e}}_{1,k}^{2} + {\hat{e}}_{2,k}^{2} + {\hat{e}}_{3,k}^{2}} \right\rbrack} = {{E\left\lbrack {\hat{e}}_{0,k}^{2} \right\rbrack} + \left\lbrack {\hat{e}}_{1,k}^{2} \right\rbrack + \left\lbrack {\hat{e}}_{2,k}^{2} \right\rbrack + \left\lbrack {\hat{e}}_{3,k}^{2} \right\rbrack}} \\{= {{\hat{\sigma}}_{e_{0}}^{2} + {\hat{\sigma}}_{e_{1}}^{2} + {\hat{\sigma}}_{e_{2}}^{2} + {\hat{\sigma}}_{e_{3}}^{2}}} \\{{= {{trace}\left\{ P_{e_{k}} \right\}}},}\end{matrix}$where {circumflex over (σ)}_(e) _(i) ² is the variance of the ithcomponent of the quaternion vector and P_(e) _(k) is the sub-block ofthe estimated state covariance matrix that relates to the quaternionsub-vector. In other wordsP _(e) _(k) =E[(e _(k) −ê _(k))(e _(k) −ê _(k))^(T)],where the indexes of the sub-vector estimate (and its covariance)relative to the full state vector estimate and covariance are given byx=[p v e b_(w)b_(a)]^(T). Based on this result, the quaternion normpseudo-observation augmentation of the observation model can beimplemented without the need to propagate extra sigma-points for thecalculation of the predicted observation and its covariance. Thesevalues already exist encapsulated and pre-calculated within theprediction of the state and its covariance. This in turn implies thatthis SPKF-centric quaternion norm regularization method can beimplemented at almost no extra computational cost.Experimental Results

The UAV platform is an R/C X-Cell 90 helicopter with custom avionics:Flight computer (300 MHz DSP400), Ashtech-G12 GPS (non-differential),Inertial Sciences ISIS-IMU, Honeywell Barometric altimeter, wirelessEthernet link, and custom servo board. However, to accurately comparethe state estimation performance of the SPKF based navigation filter toa state-of-the-art “hand tuned” EKF solution, comparative studies weremade using a high-fidelity vehicle simulator. Using a simulationenvironment for testing allows for repeatable and well controlledcomparative experiments. This simulator (designed by MIT) is builtaround a quaternion based fully nonlinear model of small helicopterdynamics utilizing a 43 dimensional state vector and accounts for mostforces and moments acting upon the vehicle. All actuators are alsomodeled. The model is parameterized by 70 discrete parameters, fit tothe true dynamics of vehicle [Gavrilets, V., Mettler, B., and Feron, E.,“Nonlinear Model for a Small-Size Acrobatic Helicopter,” in Proceedingsof AIAA Guidance Navigation and Control Conference, Montreal, Canada(August 2001)]. The vehicle control system makes use of a robust statedependent Ricatti-equation (SDRE) controller [Bogdanov, A., Wan, E. A.,Carlsson, M., Zhang, Y., Kieburtz, R., and Baptista, A., “ModelPredictive Neural Control of a High Fidelity Helicopter Model,” inProceedings of AIAA Guidance Navigation and Control Conference (2001)].

Experiment 1: In this experiment, the helicopter was flown in simulationalong a complex trajectory that increased in “aggressiveness” over time(e.g., rapid-rise-and-hover, figure-eights, and split-s). A graphicalrendering of this trajectory is shown in FIG. 5. The SDRE controllerused the true known states of the vehicle for the online calculation ofthe control law, i.e., the SPKF or EKF estimated states were not fedback to the control system. This was done to ensure that the helicopterflew exactly the same flight profile when comparing the estimationperformance of the different estimators, and is a main motivation forwhy the high-fidelity simulation environment is so attractive whencomparing different estimation approaches.

Average RMS Error position velocity Euler angles (degrees) Algorithm (m)(m/s) roll pitch Yaw EKF 2.1 0.57 0.25 0.32 2.29 SPKF - without LC 1.9(10%) 0.52 (9%) 0.20 (20%) 0.26 (19%) 1.03 (55%) SPKF - with LC 1.4(32%) 0.38 (34%) 0.17 (32%) 0.21 (34%) 0.80 (65%)

The table above summarizes the average root-mean-square (RMS) estimationerrors for the three different state estimators (EKF, SPKF, latencycompensated SPKF). The relative error reduction percentage for each ofthe two SPKF estimators compared to the EKF is shown in parentheses. Theresults show that, even though the normal SPKF already outperforms theEKF (as expected), correct accounting for GPS latency is well worth theextra effort. The plots of FIG. 6 show in graphical format the sameresults as those presented in the table (only the latency compensatedSPKF is shown relative to the EKF). FIG. 6 shows both the averageimprovement of the SPKF, as well as reduction in maximum error (as seenin “spikes” for the EKF) where the start of an aggressive maneuverexcites more nonlinear vehicle dynamics.

Experiment 2: In this experiment “closing the loop” in the GNC system isaccomplished by feeding the estimated states back to the SDRE controlsystem. In other words, the vehicle control commands will now be afunction of the estimates generated by either the EKF or SPKF estimatorand not of the “true” vehicle states. This mimics (in simulation) thetrue interdependency between the estimation and control system as wouldoccur in the real flight hardware during a fully autonomous flight. Thisexperiment thus not only indicates the difference in estimationperformance between the different filters, but also how that translatesinto improved or worsened control-performance. The average linearquadratic (LQ) control cost (J_(LQ)), which is a function of thedifference between the desired and achieved state trajectories, iscalculated as a measure of control performance.

For this experiment, the helicopter was commanded to perform anaggressive high speed nose-in turn. This maneuver requires thehelicopter to fly along an imaginary circular trajectory whileconstantly pointing its nose towards the exact center of the circle.Accurate position, velocity, and especially yaw angle estimates areneeded to accurately follow the desired flight plan with the desiredattitude. FIG. 7 shows the results of this experiment for both the EKFand SPKF. The desired flight trajectory is indicated by the red curve,the true realized trajectory in blue and the estimated trajectory ingreen. The true attitude of the helicopter is indicated by periodicrenderings of the vehicle itself along the flight path. The left-handplot shows the results for the EKF, and the right-hand plot shows theresults for the SPKF. Clearly for the SPKF case the estimated trajectoryis not only close to the true trajectory (small estimation error), butthe true trajectory is close to the desired trajectory. This indicatesgood control performance. The EKF plots clearly shows worse performanceaccording to both these criteria. Also evident from the plots is themuch improved yaw angle tracking performance of the SPKF system comparedto the EKF system. Not only does the helicopter renderings on the leftindicate that their noses are not pointing at the true center of thedesired circle, they do not even point to the same point. The SPKFsystem, on the other hand, does much better in estimating and realizingthe correct yaw attitude for this maneuver. Finally, the average controlcost of the EKF system for this maneuver was calculated as J_(LQ)=3.30,compared to the J_(LQ)=1.82 of the SPKF based system. This correspondsto a 45% reduction in control cost. These results again confirm thesuperior performance of the SPKF over the EKF.

Experiment 3: In this experiment, preliminary results in simulationcompare the robustness of the SPKF to IMU degradation. FIG. 8 shows RMSEperformance versus increase in IMU additive noise (standard scaling ofadditive white Gaussian noise). Highlighted are the nominal performancesof an Inertial Sciences ISIS-IMU (cost>10 k, weight=250 g) and the CloudCap Crista IMU (cost<2 k, weight=19 g). The Cloud Cap Crista IMU usesMEMs based gyroscopes and accelerometers manufactured by Analog Devises,and have roughly 40 times as much sensor noise as the ISIS-IMU. Asindicated by the curves, the performance of the SPKF with the cheaperand less accurate IMU is still superior to the EKF used with the moreexpensive IMU.

Actual flight data experiments: FIG. 9 shows the estimation results ofthe SPKF compared to the EKF based system on actual test flighttelemetry. The UAV was flown under pilot guidance along a complexsweeping S-shaped maneuver until it reached a specified altitude (50 m),at which point the system was switched to fully autonomous flight. Theautonomous flight plan entailed holding the UAV steady in hover for anumber of seconds, flying the UAV in a square trajectory at a constantaltitude of about 55-60 meters, and again holding the UAV steady inhover for a number of seconds before landing. Because there was noground truth signal available for absolute error comparison the resultsare evaluated on more subjective terms. A top-down (2D) projection ofthe estimation results, which is shown in FIG. 10, provides good insightfor this purpose. FIG. 10 shows that a significant number of GPS outagesoccurred during the pilot guided ascent to the hovering altitude(S-shaped curve). Clearly, the SPKF appears to more accurately track the(assumed) true underlying trajectory during this outage period.Coordinates {40-60} of FIG. 10 indicate that the EKF generated positionestimate exhibits an erratic jump just before the GPS measurements againbecome available. FIGS. 11A, 11B, and 11C show the effect of the GPSoutages on, respectively, the separate north, east, and down componentsof the estimated 3D positions. This error results from the inherentnature of the INS solution (derived from integrating the biascompensated IMU gyroscope and accelerometer data) to drift duringperiods of GPS outage. Since the SPKF performs a more accuratetime-update during these outage periods and possibly more accuratelytracks the underlying IMU biases than does the EKF, the resulting SPKFestimates are more robust to GPS outages in general.

Although the above-described preferred embodiment implemented a directformulation of the kinematics equations, an alternative preferredembodiment of the invention, implements an indirect (error) formulation.In the error formulation, the Kalman filter estimates the errors in thenavigation and attitude information. The state variables of the indirectformulation are not position, velocity, and attitude but are the errorsin an Inertial Navigation System (INS) computation of position,velocity, and attitude. (The INS uses the direct form of the kinematicequations.) The system observation in the indirect error stateformulation is the difference between INS computation and GPSmeasurements. The error state Kalman filter generates estimates of theerrors in the inertial system using external observations that may beupdated at a rate much lower than the INS computation rate. This is incontrast to the direct form implementation, which needs to update thefilter at the INS rate (e.g., the IMU rate). After each update of theerror state Kalman filter with new GPS measurements, the INS iscorrected by feedback of the error estimates and the predicted errorstate is set to zero for the next sample time.

More specifically, analysis of the indirect formulation starts with thestandard Kinematic model used within the INS (inertial navigationsystem),

$\begin{matrix}{{\frac{\mathbb{d}}{\mathbb{d}t}p} = v} \\{{\frac{\mathbb{d}}{\mathbb{d}t}v} = {{C_{b}^{n}f_{b}} + g}} \\{{{\frac{\mathbb{d}}{\mathbb{d}t}C_{b}^{n}} = {C_{b}^{n}\left( {w_{nb}^{b} \times} \right)}},}\end{matrix}$where C_(b) ^(n) is the Direction Cosine Matrix (DCM) from the bodyframe to the navigation frame, i.e., the local NED frame. The angularrate vector of the body frame with respect to the navigation frameprojected to the body frame is given by the skew-symmetric matrix ofW_(nb) ^(b),

$\left( {w_{nb}^{b} \times} \right) = {\begin{pmatrix}0 & {- w_{z}} & w_{y} \\w_{z} & 0 & {- w_{x}} \\{- w_{y}} & w_{x} & 0\end{pmatrix}.}$

The position and velocity measurement equation for the GPS, taking intoaccount the lever-arm effect isp _(GPS) =p+C _(b) ^(n) r _(GPS) +n _(GPS) ^(p)v _(GPS) =v+C _(b) ^(n) w _(nb) ^(b) ×r _(GPS) +n _(GPS) ^(v),  (14)where r_(GPS) is the position of the GPS sensors in the body frame andn_(GPS) is the noise of GPS measurements.

The differences between the INS computations of position, velocity, andquaternion and their true values areδp={tilde over (p)}−pδv={tilde over (v)}−vδq={tilde over (q)} ⁻¹ ·q,where ‘˜’ denotes values computed by the INS, and ‘.’ denotes quaternioncomposition. The quaternion can be decomposed as a scalar and a 3×1vector

${\delta\; q} = {\left( \frac{\delta\; q_{0}}{\delta\; q} \right).}$

The dynamics for the error states are then given as:

-   -   1) Position error

$\begin{matrix}{{\frac{\mathbb{d}}{\mathbb{d}t}\delta\; p} = {\delta\; v}} & (15)\end{matrix}$

-   -   2) Attitude error (represented in quaternion)

$\begin{matrix}{{\frac{\mathbb{d}}{\mathbb{d}t}\overset{\rightarrow}{\delta\; q}} \approx {{{- {\overset{\sim}{w}}_{nb}^{b}} \times \overset{\rightarrow}{\delta\; q}} - {0.5\;\delta\; w_{nb}^{b}}}} & (16)\end{matrix}$

-   -   3) Velocity error

$\begin{matrix}{{\frac{\mathbb{d}}{\mathbb{d}t}\delta\; v} \approx {{\left( {2\;{{\overset{\sim}{C}}_{b}^{n}\left( {{\overset{\sim}{f}}_{b} \times} \right)}} \right)\mspace{11mu}\overset{\rightarrow}{\delta\; q}} + {{\overset{\sim}{C}}_{b}^{n}\;\delta\; f_{b}}}} & (17)\end{matrix}$Because the navigation range of the vehicle is small, the relativemotion between the navigation frame and the inertial frame is neglectedin the attitude error expression. Approximations stem from neglectingcross-terms of the error states and the measurement uncertainty. The newerror state vector is defined as

$X = {\begin{pmatrix}{\delta\; p} \\{\delta\; v} \\\overset{\rightarrow}{\delta\; q}\end{pmatrix}.}$

The position, attitude, and velocity error state Expressions (6-8)provide

$\begin{matrix}{{\frac{\mathbb{d}}{\mathbb{d}t}X} = {{\begin{pmatrix}0 & I_{3} & 0 \\0 & 0 & {2\;{{\overset{\sim}{C}}_{b}^{n}\left( {\overset{\sim}{f}}_{b} \right)}} \\0 & 0 & {- \left( {{\overset{\sim}{w}}_{nb}^{b} \times} \right)}\end{pmatrix}X} + {\begin{pmatrix}0 & 0 \\0 & {\overset{\sim}{C}}_{b}^{n} \\{{- 0.5}\; I_{3}} & 0\end{pmatrix}{\begin{pmatrix}{\delta\; w_{nb}^{b}} \\{\delta\; f_{b}}\end{pmatrix}.}}}} & (18)\end{matrix}$

To specify the observation equation, one may write the estimation of theGPS measurements using the INS computation as{tilde over (p)} _(GPS) ={tilde over (p)}+{tilde over (C)} _(b) ^(u) r_(GPS){tilde over (v)} _(GPS) ={tilde over (v)}+{tilde over (C)} _(b) ^(n){tilde over (w)} _(nb) ^(b) ×r _(GPS)  (19)

Subtracting Equation (14) from Equation (19), and using the definitionof the error states, provides

$\begin{matrix}\begin{matrix}{y = {\begin{pmatrix}{\overset{\sim}{p}}_{GPS} \\{\overset{\sim}{v}}_{GPS}\end{pmatrix} - \begin{pmatrix}p_{GPS} \\v_{GPS}\end{pmatrix}}} \\{\approx {{\begin{pmatrix}I_{3} & 0 & {2\;{{\overset{\sim}{C}}_{b}^{n}\left( {r_{GPS} \times} \right)}} \\0 & I_{3} & {2\;{{\overset{\sim}{C}}_{b}^{n}\left( {\left( {{\overset{\sim}{w}}_{nb}^{b} \times r_{GPS}} \right) \times} \right)}}\end{pmatrix}X} - {\begin{pmatrix}n^{p} \\n^{v}\end{pmatrix}_{GPS}.}}}\end{matrix} & (20)\end{matrix}$

Equation (18) and Equation (20) describe the error state dynamics andthe state observation, respectively. The EKF or SPKF can then be appliedto these system equations. Alternative forms of the error state-spacemay be obtained by either using Euler angles (instead of quaternions) orusing the error of the DCM as a state.

The preferred embodiment was described with reference to a looselycoupled GPS/INS integration for UAV autonomy. The invention can also beapplied to a tightly coupled integration, in which raw satellite datasignal transmissions (e.g., pseudo-range code, carrier phase, andDoppler) are integrated directly with the IMU sensors, therebyeffectively bypassing the GPS receiver. The observation data from IMUand raw satellite signal transmissions from the GPS contribute to theprobabilistic inference system to enable complete estimation of thenavigational state of the system, IMU specific ancillary parameters, andGPS specific ancillary parameters.

The invention can also be applied to only the GPS to integrate rawsatellite transmissions in accordance with an SPKF-based approach incontrast to an EKF-based approach. In such an embodiment, the rawsatellite signal transmissions are used to estimate a set ofnavigational state components, including but not limited to, positionand velocity information relating to the GPS receiver.

Although the preferred embodiment was described for an SPKF basedprobabilistic inference system implemented with the square-root versionof CDKF (SR-CDKF) algorithm, other sigma-point approach algorithms canbe used. A list of examples of such algorithms includes unscented Kalmanfilter (UKF) algorithms (Algorithm 1), central difference Kalman filter(CDKF) algorithms (Algorithm 2), square-root version of UKF (SR-UKF)(Algorithm 3), square-root version of CDKF (SR-CDKF) (Algorithm 4),sigma-point particle filter (SPPF) algorithm (Algorithm 5), and Gaussianmixture sigma-point particle filter (GMSPPF) algorithm (Algorithm 6).Each parenthetical reference identifies one of the particular algorithmspresented below as a collection of the mathematical algorithms listedabove.

Algorithm 1: The Unscented Kalman Filter (UKF)

Initialization: {circumflex over (x)}₀=E[x₀, P_(x) ₀ =E[(x₀−{circumflexover (x)}₀)(x₀−{circumflex over (x)}₀)^(T)]

$\begin{matrix}{{{\hat{x}}_{0}^{a} = {{E\left\lbrack x^{a} \right\rbrack} = {E\begin{bmatrix}{\hat{x}}_{0} & 0 & 0\end{bmatrix}}^{T}}},} \\{P_{0}^{a} = {{E\left\lbrack {\left( {x_{0}^{a} - {\hat{x}}_{0}^{a}} \right)\mspace{11mu}\left( {x_{0}^{a} - {\hat{x}}_{0}^{a}} \right)^{T}} \right\rbrack} = \begin{bmatrix}P_{x_{0}} & 0 & 0 \\0 & R_{v} & 0 \\0 & 0 & R_{n}\end{bmatrix}}}\end{matrix}$

For k=1, . . . , ∞:

-   -   1. Calculate sigma-points:        χ_(k−1) ^(a)=[{circumflex over (x)}_(k−1) ^(a){circumflex over        (x)}_(k−1) ^(a)+γ√{square root over (P _(k−1) ^(a))}{circumflex        over (x)}_(k−1) ^(a)−γ√{square root over (P _(k−1) ^(a))}]    -   2. Time-update equations:

$\begin{matrix}{{??}_{k|{k - 1}}^{x} = {f\left( {{??}_{k - 1}^{x},{??}_{k - 1}^{v},u_{k - 1}} \right)}} \\{{\hat{x}}_{k}^{-} = {\sum\limits_{i = 0}^{2L}{w_{i}^{(m)}\;{??}_{i,{k|{k - 1}}}^{x}}}} \\{P_{x_{k}}^{-} = {\sum\limits_{i = 0}^{2L}{{w_{i}^{(c)}\left( {{??}_{i,{k|{k - 1}}}^{x} - {\hat{x}}_{k}^{-}} \right)}\;\left( {{??}_{i,{k|{k - 1}}}^{x} - {\hat{x}}_{k}^{-}} \right)^{T}}}}\end{matrix}$

-   -   3. Measurement-update equations:

$\begin{matrix}{{??}_{k|{k - 1}} = {h\left( {{??}_{k|{k - 1}}^{x},{??}_{k - 1}^{n}} \right)}} \\{{\hat{y}}_{k}^{-} = {\sum\limits_{i = 0}^{2L}{w_{i}^{(m)}{??}_{i,{k|{k - 1}}}}}} \\{P_{{\overset{\sim}{y}}_{k}} = {\sum\limits_{i = 0}^{2L}{{w_{i}^{(c)}\left( {{??}_{i,{k|{k - 1}}} - {\hat{y}}_{k}^{-}} \right)}\left( {{??}_{i,{k|{k - 1}}} - {\hat{y}}_{k}^{-}} \right)^{T}}}} \\{P_{x_{k}y_{k}} = {\sum\limits_{i = 0}^{2L}{{w_{i}^{(c)}\left( {{??}_{i,{k|{k - 1}}}^{x} - {\hat{x}}_{k}^{-}} \right)}\left( {{??}_{i,{k|{k - 1}}} - {\hat{y}}_{k}^{-}} \right)^{T}}}} \\{K_{k} = {P_{x_{k}y_{k}}P_{{\overset{\sim}{y}}_{k}}^{- 1}}} \\{{\hat{x}}_{k} = {{\hat{x}}_{k}^{-} + {K_{k}\left( {y_{k} - {\hat{y}}_{k}^{-}} \right)}}} \\{P_{x_{k}} = {P_{x_{k}}^{-} - {K_{k}P_{{\overset{\sim}{y}}_{k}}K_{k}^{T}}}}\end{matrix}$

Parameters: x^(a)=x^(T)v^(T)n^(T)]^(T),χ^(a)=[(χ^(x))^(T)(χ^(v))^(T)(χ^(n))^(T)]^(T), γ=√{square root over(L+γ)}: γ is a composite scaling parameter and λ is given in [0007], Lis the dimension of the augmented states, R_(v) is the process-noisecovariance, R_(n) is the observation-noise covariance, and w_(i) are theweights as calculated in [0007].

Algorithm 2: The Central Difference Kalman Filter (CDKF)—SPKFFormulation

Initialization: {circumflex over (x)}₀=E[x₀], p_(x) ₀ =E[(x₀−{circumflexover (x)}₀)(x₀−{circumflex over (x)}₀)^(T)]

For k=1, . . . , ∞:

-   -   1. Calculate sigma-points for time-update:

$\begin{matrix}\begin{matrix}{{{\hat{x}}_{k - 1}^{a_{v}} = \begin{bmatrix}{\hat{x}}_{k - 1} & \overset{\_}{v}\end{bmatrix}},} & {P_{k - 1}^{a_{v}} = \begin{bmatrix}P_{x_{k - 1}} & 0 \\0 & R_{v}\end{bmatrix}}\end{matrix} \\{{??}_{k - 1}^{a_{v}} = \begin{bmatrix}{\hat{x}}_{k - 1}^{a_{v}} & {{\hat{x}}_{k - 1}^{a_{v}} + {h\sqrt{P_{k - 1}^{a_{v}}}}} & {{\hat{x}}_{k - 1}^{a_{v}} - {h\sqrt{P_{k - 1}^{a_{v}}}}}\end{bmatrix}}\end{matrix}$

-   -   2. Time-update equations:

??_(k|k − 1)^(x) = f (??_(k − 1)^(x), ??_(k − 1)^(v), u_(k − 1))${\hat{x}}_{k}^{-} = {\sum\limits_{i = 0}^{2L}{w_{i}^{(m)}\;{??}_{i,{k|{k - 1}}}^{x}}}$$P_{x_{k}}^{-} = {\sum\limits_{i = 1}^{L}\left\lbrack {{w_{i}^{(c_{1})}\left( {{??}_{i,{k|{k - 1}}}^{x} - {??}_{{L + i},{k|{k - 1}}}^{x}} \right)}^{2} + {w_{i}^{(c_{2})}\left( {{??}_{i,{k|{k - 1}}}^{x} + {??}_{{L + i},{k|{k - 1}}}^{x} - {2\;{??}_{0,{k|{k - 1}}}^{x}}} \right)}^{2}} \right\rbrack}$

-   -   3. Calculate sigma-points for measurement-update:

$\begin{matrix}\begin{matrix}{{{\hat{x}}_{k|{k - 1}}^{a_{n}} = \begin{bmatrix}{\hat{x}}_{k}^{-} & \overset{\_}{n}\end{bmatrix}},} & {P_{k|{k - 1}}^{a_{n}} = \begin{bmatrix}P_{x_{k}}^{-} & 0 \\0 & R_{n}\end{bmatrix}}\end{matrix} \\{{??}_{k|{k - 1}}^{a_{n}} = \begin{bmatrix}{\hat{x}}_{k|{k - 1}}^{a_{n}} & {{\hat{x}}_{k|{k - 1}}^{a_{n}} + {h\sqrt{P_{k|{k - 1}}^{a_{n}}}}} & {{\hat{x}}_{k|{k - 1}}^{a_{n}} - {h\sqrt{P_{k|{k - 1}}^{a_{n}}}}}\end{bmatrix}}\end{matrix}$

-   -   4. Measurement-update equations:

  ??_(k|k − 1) = h(??_(k|k − 1)^(x), ??_(k|k − 1)^(n))$\mspace{20mu}{{\hat{y}}_{k}^{-} = {\sum\limits_{i = 0}^{2L}{w_{i}^{(m)}\;{??}_{i,{k|{k - 1}}}}}}$$P_{{\overset{\sim}{y}}_{k}} = {\sum\limits_{i = 1}^{L}\left\lbrack {{w_{i}^{(c_{1})}\;\left( {{??}_{i,{k|{k - 1}}} - {??}_{{L + i},{k|{k - 1}}}} \right)^{2}} + {w_{i}^{(c_{2})}\;\left( {{??}_{i,{k|{k - 1}}} + {??}_{{L + i},{k|{k - 1}}} - {2\;{??}_{0,{k|{k - 1}}}}} \right)^{2}}} \right\rbrack}$$\mspace{20mu}{P_{x_{k}y_{k}} = \;{= {\sqrt{w_{1}^{(c_{1})}P_{x_{k}}^{-}}\;\left\lbrack {{??}_{{1:L},{k|{k - 1}}} - {??}_{{{L + 1}:{2L}},{k|{k - 1}}}} \right\rbrack}^{T}}}$$\mspace{20mu}{K_{k} = {P_{x_{k}y_{k}}P_{{\overset{\sim}{y}}_{k}}^{- 1}}}$  x̂_(k) = x̂_(k)⁻ + K_(k)(y_(k) − ŷ_(k)⁻)$\mspace{20mu}{P_{x_{k}} = {P_{x_{k}}^{-} - {K_{k}P_{{\overset{\sim}{y}}_{k}}K_{k}^{T}}}}$

Parameters: x^(a) ^(v) =[x^(T)v^(T)]^(T), χ^(a) ^(v)=[(χ^(x))^(T)(χ^(v))^(T)]^(T), x^(an)=[x^(T) n^(T)]^(T),χ^(an)=[(χ^(x))^(T)(χ^(n))^(T)]^(T), h≧1 is the scalar centraldifference step size, L is the dimension of the augmented states, R_(v)is the process-noise covariance, R_(n) is the observation-noisecovariance, and w_(i) are the weights as calculated in Equation (**).(·)² is shorthand for the vector outer product, i.e. a²{dot over(=)}aa^(T).

General note: Here we again augment the system state with the processnoise and observation noise vectors (v_(k) and n_(k)) as we did for theUKF. For the CDKF, however, we split this augmentation between thetime-update and measurement-update, i.e., for the time-update theaugmented state vector and augmented covariance matrix is given by

$\begin{matrix}{{x_{k}^{a_{v}} = \begin{bmatrix}x_{k}^{T} & v_{k}^{T}\end{bmatrix}^{T}},} & {{P_{k}^{a_{v}} = \begin{bmatrix}P_{x_{k}} & 0 \\0 & R_{v}\end{bmatrix}},}\end{matrix}$and by

$\begin{matrix}{{x_{k}^{a_{n}} = \begin{bmatrix}x_{k}^{T} & n_{k}^{T}\end{bmatrix}^{T}},} & {{P_{k}^{a_{n}} = \begin{bmatrix}P_{x_{k}} & 0 \\0 & R_{n}\end{bmatrix}},}\end{matrix}$for the measurement-update.

$\begin{matrix}{{{??}_{0} = \overset{\_}{x}}\begin{matrix}{{??}_{i} = {\overset{\_}{x} + \left( {h\;\sqrt{P_{x}}} \right)_{i}}} & {{i = 1},\ldots\mspace{11mu},L}\end{matrix}\begin{matrix}{{??}_{i} = {\overset{\_}{x} - \left( {h\;\sqrt{P_{x}}} \right)_{i}}} & {{i = {L + 1}},\ldots\mspace{11mu},{2L}}\end{matrix}{w_{0}^{(m)} = \frac{h^{2} - L}{h^{2}}}\begin{matrix}{w_{i}^{(m)} = \frac{1}{2h^{2}}} & \; & {\mspace{11mu}{{i = 1},\ldots\mspace{11mu},{2L}}}\end{matrix}\begin{matrix}{w_{i}^{(c_{1})} = \frac{1}{4h^{2}}} & \; & {{i = 1},\ldots\mspace{11mu},{2L}}\end{matrix}\begin{matrix}{w_{i}^{(c_{2})} = \frac{h^{2} - 1}{4h^{4}}} & {{i = 1},\ldots\mspace{11mu},{2L}}\end{matrix}} & {{(*}{*)}}\end{matrix}$Algorithm 3: The Square-Root UKF (SR-UKF)—General State Estimation Form

-   -   Initialization: {circumflex over (x)}₀=E[x₀], S_(x) ₀        =chol{E[(x₀−{circumflex over (x)}₀)(x₀−{circumflex over        (x)}₀)^(T)]}, S_(v)=√{square root over (R_(v))}, S_(n)=√{square        root over (R_(v))}

$\begin{matrix}{{{\hat{x}}_{0}^{a} = {{E\left\lbrack x^{a} \right\rbrack} = \begin{bmatrix}{\hat{x}}_{0} & 0 & 0\end{bmatrix}^{T}}},} \\{S_{0}^{a} = {{{chol}\left\{ {E\left\lbrack {\left( {x_{0}^{a} - {\hat{x}}_{0}^{a}} \right)\left( {x_{0}^{a} - {\hat{x}}_{0}^{a}} \right)^{T}} \right\rbrack} \right\}} = \begin{bmatrix}S_{x_{0}} & 0 & 0 \\0 & S_{v} & 0 \\0 & 0 & S_{n}\end{bmatrix}}}\end{matrix}$

For k=1, . . . , ∞:

-   -   1. Calculate sigma-points:        χ_(k−1) ^(a) =[{circumflex over (x)} _(k−1) ^(a) {circumflex        over (x)} _(k−1) ^(a) +γS _(x) _(k−1) ^(a) {circumflex over (x)}        _(k−1) −65 S _(xk−1) ^(a) {circumflex over (x)} _(k−1) ^(a) −γS        _(xk−1) ^(a)]    -   2. Time-update equations:

$\begin{matrix}{{??}_{k|{k - 1}}^{x} = {f\left( {{??}_{k - 1}^{a},{??}_{k - 1}^{v},u_{k - 1}} \right)}} \\{{\hat{x}}_{k}^{-} = {\sum\limits_{i = 0}^{2L}{w_{i}^{(m)}\;{??}_{i,{k|{k - 1}}}^{x}}}} \\{S_{x_{k}}^{-} = {{qr}\;\left\{ \left\lbrack {\sqrt{w_{1}^{(c)}}\left( {{??}_{{1:{2L}},{k|{k - 1}}}^{x} - {\hat{x}}_{k}^{-}} \right)} \right\rbrack \right\}}} \\{S_{x_{k}}^{-} = {{cholupdate}\;\left\{ {S_{x_{k}}^{-},{{??}_{0,{k|{k - 1}}}^{x} - {\hat{x}}_{k}^{-}},w_{0}^{(c)}} \right\}}} \\{{??}_{k|{k - 1}} = {h\;\left( {{??}_{i,{k|{k - 1}}}^{x},{??}_{k - 1}^{n}} \right)}} \\{{\hat{y}}_{k}^{-} = {\sum\limits_{i = 0}^{2L}{w_{i}^{(m)}\;{??}_{i,{k|{k - 1}}}}}}\end{matrix}$

-   -   3. Measurement-update equations:

$\begin{matrix}{S_{{\overset{\sim}{y}}_{k}} = {{qr}\left\{ \left\lbrack {\sqrt{w_{1}^{(c)}}\left( {{??}_{{1:{2L}},{k|{k - 1}}} - {\hat{y}}_{k}^{-}} \right)} \right\rbrack \right\}}} \\{S_{{\overset{\sim}{y}}_{k}} = {{cholupdate}\;\left\{ {S_{{\overset{\sim}{y}}_{k}},{{??}_{0,{k|{k - 1}}} - {\hat{y}}_{k}^{-}},w_{0}^{(c)}} \right\}}} \\{P_{x_{k}y_{k}} = {\sum\limits_{i = 0}^{2L}{w_{i}^{(c)}\;\left( {{??}_{i,{k|{k - 1}}}^{x} - {\hat{x}}_{k}^{-}} \right)\mspace{11mu}\left( {{??}_{i,{k|{k - 1}}} - {\hat{y}}_{k}^{-}} \right)^{T}}}} \\{K_{k} = {\left( {P_{x_{k}y_{k}}/S_{{\overset{\sim}{y}}_{k}}^{T}} \right)/S_{{\overset{\sim}{y}}_{k}}}} \\{{\hat{x}}_{k} = {{\hat{x}}_{k}^{-} + {K_{k}\left( {y_{k} - {\hat{y}}_{k}^{-}} \right)}}} \\{U = {K_{k}\; S_{{\overset{\sim}{y}}_{k}}}} \\{S_{x_{k}} = {{cholupdate}\;\left\{ {S_{x_{k}}^{-},U,{- 1}} \right\}}}\end{matrix}$

Parameters: x^(a)=[x^(T)v^(T)n^(T)]^(T),χ^(a)=[(χ^(x))^(T)(χ^(v))^(T)(χ^(n))^(T)], γ=√{square root over (L+λ)}isthe composite scaling parameter and λ is given in [0007], L is thedimension of the state, R_(v) is the process-noise covariance, R_(n) isthe observation-noise covariance, and w_(i) are the weights ascalculated in [0007].

Algorithm 4: The Square-Root CDKF (SR-CDKF)—General State EstimationForm

Initialization: {circumflex over (x)}₀=E[x₀],S_(x0)=chol{E[(x₀−{circumflex over (x)}₀)(x₀−{circumflex over(x)}₀)^(T)]}, S_(v)=√{square root over (R_(v))}, S_(n=√){square rootover (R_(v))}

For k=1, . . . , ∞:

-   -   1. Calculate sigma points for time-update:

$\begin{matrix}\begin{matrix}{{{\hat{x}}_{k - 1}^{a_{v}} = \begin{bmatrix}{\hat{x}}_{k - 1} & \overset{\_}{v}\end{bmatrix}},} & {S_{k - 1}^{a_{v}} = \begin{bmatrix}S_{x_{k - 1}} & 0 \\0 & S_{v}\end{bmatrix}}\end{matrix} \\{{??}_{k - 1}^{a_{v}} = \begin{bmatrix}{\hat{x}}_{k - 1}^{a_{v}} & {{\hat{x}}_{k - 1}^{a_{v}} + {h\; S_{k - 1}^{a_{v}}}} & {{\hat{x}}_{k - 1}^{a_{v}} - {h\; S_{k - 1}^{a_{v}}}}\end{bmatrix}}\end{matrix}$

-   -   2. Time-update equations:

  ??_(k|k − 1)^(x) = f (??_(k − 1)^(x), ??_(k − 1)^(v), u_(k − 1))$\mspace{20mu}{{\hat{x}}_{k}^{-} = {\sum\limits_{i = 0}^{2L}{w_{i}^{(m)}\;{??}_{i,{k|{k - 1}}}^{x}}}}$$S_{x_{k}}^{-} = {{qr}\;\left\{ \left\lbrack {\sqrt{w_{1}^{(c_{1})}}\;\left( {{??}_{{1:L},{k|{k - 1}}}^{x} - {??}_{{{L + 1}:{2L}},{k|{k - 1}}}^{x}} \right)\;\sqrt{w_{1}^{(c_{2})}}\;\left( {{??}_{{1:L},{k|{k - 1}}}^{x} + {??}_{{{L + 1}:{2L}},{k|{k - 1}}}^{x} - {2\;{??}_{0,{k|{k - 1}}}^{x}}} \right)} \right\rbrack \right\}}$

-   -   3. Calculate sigma-points for measurement update:

$\begin{matrix}\begin{matrix}{{{\hat{x}}_{k|{k - 1}}^{a_{n}} = \begin{bmatrix}{\hat{x}}_{k}^{-} & \overset{\_}{n}\end{bmatrix}},} & {S_{k|{k - 1}}^{a_{n}} = \begin{bmatrix}S_{x_{k}}^{-} & 0 \\0 & S_{n}\end{bmatrix}}\end{matrix} \\{{??}_{k|{k - 1}}^{a_{n}} = \begin{bmatrix}{\hat{x}}_{k|{k - 1}}^{a_{n}} & {{\hat{x}}_{k|{k - 1}}^{a_{n}} + {h\; S_{k|{k - 1}}^{a_{n}}}} & {{\hat{x}}_{k|{k - 1}}^{a_{n}} - {h\; S_{k|{k - 1}}^{a_{n}}}}\end{bmatrix}}\end{matrix}$

-   -   4. Measurement-update equations:

  ??_(k|k − 1) = h (??_(k|k − 1)^(x), ??_(k|k − 1)^(n))$\mspace{20mu}{{\hat{y}}_{k}^{-} = {\sum\limits_{i = 0}^{2L}{w_{i}^{(m)}\;{??}_{i,{k|{k - 1}}}}}}$$S_{{\overset{\sim}{y}}_{k}} = {{qr}\;\left\{ \left\lbrack {\sqrt{w_{1}^{(c_{1})}}\;\left( {{??}_{{1:L},{k|{k - 1}}} - {??}_{{{L + 1}:{2L}},{k|{k - 1}}}} \right)\;\sqrt{w_{1}^{(c_{2})}}\;\left( {{??}_{{1:L},{k|{k - 1}}} - {??}_{{{L + 1}:{2L}},{k|{k - 1}}} - {2\;{??}_{0,{k|{k - 1}}}}} \right)} \right\rbrack \right\}}$P _(xkyk)=√{square root over (w ₁ ^((c) ¹ ⁾)}S _(xk) [y _(1:L,k|k−1) −y_(L+1:2L,k|k−1)]⁹K _(k)=(P _(xkyk) /S _({tilde over (y)}k) ^(T))/S _({tilde over (y)}k){circumflex over (x)} _(k) ={circumflex over (x)}_(k) +K _(k)(y _(k) −ŷ_(k) )U=K_(k)S _(yk)S _(xk)=cholupdate{S _(xk) , U, −1}

Parameters: x^(av)=[x^(T) v^(T)]^(T),χ^(av)=[(χ^(x))^(T)(χ^(v))^(T)]^(T), x^(an)=[x^(T) n^(T)]^(T),χ^(an)=[(χ^(x))^(T)(χ^(n))^(T)]^(T), h≧1 is the scalar centraldifference step size, L is the dimension of the augmented states, R_(v)is the process-noise covariance R_(n) is the observation-noisecovariance, and w_(i) are the weights as calculated in Equation (**)(·)² is shorthand for the vector outer product.

$\begin{matrix}{{{??}_{0} = \overset{\_}{x}}\begin{matrix}{{??}_{i} = {\overset{\_}{x} + \left( {h\;\sqrt{P_{x}}} \right)_{i}}} & {{i = 1},\ldots\mspace{11mu},L}\end{matrix}\begin{matrix}{{??}_{i} = {\overset{\_}{x} - \left( {h\;\sqrt{P_{x}}} \right)_{i}}} & {{i = {L + 1}},\ldots\mspace{11mu},{2L}}\end{matrix}{w_{0}^{(m)} = \frac{h^{2} - L}{h^{2}}}\begin{matrix}{w_{i}^{(m)} = \frac{1}{2h^{2}}} & \; & {\mspace{11mu}{{i = 1},\ldots\mspace{11mu},{2L}}}\end{matrix}\begin{matrix}{w_{i}^{(c_{1})} = \frac{1}{4h^{2}}} & \; & {{i = 1},\ldots\mspace{11mu},{2L}}\end{matrix}\begin{matrix}{w_{i}^{(c_{2})} = \frac{h^{2} - 1}{4h^{4}}} & {{i = 1},\ldots\mspace{11mu},{2L}}\end{matrix}} & {{(*}{*)}}\end{matrix}$Algorithm 5 The Sigma-Point Particle Filter (SPPF)

The new filter that results from using a SPKF for proposal distributiongeneration within a particle filter framework is called the sigma-pointparticle filter (SPPF):

-   -   Initialization: k=0        -   1. For i=1, . . . , N, draw (sample) particle x₀ ^((i)) from            the prior p(x₀).    -   For k=1, 2, . . .        -   1. Importance sampling step    -   For i=1, . . . , N:        -   (a) Update the Gaussian prior distribution for each particle            with the SPKF        -   Calculate sigma-points for particle,

$\begin{matrix}{x_{k - 1}^{a,{(i)}} = {\begin{bmatrix}x_{k - 1}^{(i)} & {\overset{\_}{v}}_{k - 1} & {\overset{\_}{n}}_{k - 1}\end{bmatrix}^{T}\text{:}}} \\{{??}_{{k - 1},{({0\mspace{11mu}\ldots\mspace{11mu} 2L})}}^{a,{(i)}} = \begin{bmatrix}x_{k - 1}^{a,{(i)}} & {x_{k - 1}^{a,{(i)}} + {\gamma\sqrt{P_{k - 1}^{a,{(i)}}}}} & {x_{k - 1}^{a,{(i)}} - {\gamma\;\sqrt{P_{k - 1}^{a,{(i)}}}}}\end{bmatrix}}\end{matrix}$

-   -   -   -   Propagate sigma-points into future (time update):

$\begin{matrix}{{??}_{{k|{k - 1}},{({0\mspace{11mu}\ldots\mspace{11mu} 2L})}}^{x,{(i)}} = {f\;\left( {{??}_{{k - 1},{({0\mspace{11mu}\ldots\mspace{11mu} 2L})}}^{x,{(i)}},{??}_{{k - 1},{({0\mspace{11mu}\ldots\mspace{11mu} 2L})}}^{v,{(i)}},u_{k}} \right)}} \\{{\overset{\_}{x}}_{k|{k - 1}}^{(i)} = {\sum\limits_{j = 0}^{2L}{w_{j}^{(m)}\;{??}_{{k|{k - 1}},j}^{x,{(i)}}}}} \\{P_{k|{k - 1}}^{(i)} = {\sum\limits_{j = 0}^{2L}{{w_{j}^{(c)}\left( {{??}_{{k|{k - 1}},j}^{x,{(i)}} - {\overset{\_}{x}}_{k|{k - 1}}^{(i)}} \right)}\;\left( {{??}_{{k|{k - 1}},j}^{x,{(i)}} - {\overset{\_}{x}}_{k|{k - 1}}^{(i)}} \right)^{T}}}} \\{{??}_{{k|{k - 1}},{({0\mspace{11mu}\ldots\mspace{11mu} 2L})}}^{(i)} = {h\;\left( {{??}_{{k|{k - 1}},{({0\mspace{11mu}\ldots\mspace{11mu} 2L})}}^{x,{(i)}},{??}_{{k - 1},{({0\mspace{11mu}\ldots\mspace{11mu} 2L})}}^{n,{(i)}}} \right)}} \\{{\overset{\_}{y}}_{k|{k - 1}}^{(i)} = {\sum\limits_{j = 0}^{2L}{W_{j}^{(m)}\;{??}_{j,{k|{k - 1}}}^{(i)}}}}\end{matrix}$

-   -   -   -   Incorporate new observation (measurement update):

$\begin{matrix}{P_{y_{k}y_{k}} = {\sum\limits_{j = 0}^{2L}{{w_{j}^{(c)}\left\lbrack {{??}_{{k|{k - 1}},j}^{(i)} - {\overset{\_}{y}}_{k|{k - 1}}^{(i)}} \right\rbrack}\left\lbrack {{??}_{{k|{k - 1}},j}^{(i)} - {\overset{\_}{y}}_{k|{k - 1}}^{(i)}} \right\rbrack}^{T}}} \\{P_{x_{k}y_{k}} = {\sum\limits_{j = 0}^{2L}{{w_{j}^{(c)}\left\lbrack {{??}_{{k|{k - 1}},j}^{(i)} - {\overset{\_}{x}}_{k|{k - 1}}^{(i)}} \right\rbrack}\left\lbrack {{??}_{{k|{k - 1}},j}^{(i)} - {\overset{\_}{y}}_{k|{k - 1}}^{(i)}} \right\rbrack}^{T}}} \\{K_{k} = {P_{x_{k}y_{k}}P_{x_{k}y_{k}}^{- 1}}} \\{{\overset{\_}{x}}_{k}^{(i)} = {{\overset{\_}{x}}_{k|{k - 1}}^{(i)} + {K_{k}\left( {y_{k} - {\overset{\_}{y}}_{k|{k - 1}}^{(i)}} \right)}}} \\{P_{k}^{(i)} = {P_{k|{k - 1}}^{(i)} - {K_{k}P_{y_{k}y_{k}}K_{k}^{T}}}}\end{matrix}$

-   -   -   -   (b) Sample x_(k) ^((i))˜q_(N)(x_(k)|y_(1:k))=N(x_(k); x                _(k) ^((i)), P_(k) ^((i)))

    -   For i=1, . . . , N, evaluate the importance weights up to a        normalizing constant:

$w_{k}^{(i)} = {w_{k - 1}^{(i)}\frac{{p\left( y_{k} \middle| x_{k}^{(i)} \right)}\;{p\left( x_{k}^{(i)} \middle| x_{k - 1}^{(i)} \right)}}{q_{??}\left( x_{k}^{(i)} \middle| y_{1:k} \right)}}$

-   -   For i=1, . . . , N, normalize the importance weights: {tilde        over (w)}_(k) ^((i)=w) _(k) ^((i))/Σ_(j=1) ^(N)w_(k) ^((i))

2. Selection step (resampling)

-   -   Multiply/suppress samples x_(k) ^((i)) with high/low importance        weights {tilde over (w)}_(k) ^((i)), respectively, to obtain N        random samples approximately distributed according to        p(x_(k)|y_(1:k))    -   For i=1, . . . , N, set w_(k) ^((i)={tilde over (w)}) _(k)        ^((i)=N⁻¹.    -   (optional) Do a single MCMC (Markov chain Monte Carlo) move step        to add further ‘variety’ to the particle set without changing        their distribution.

3. Output: The output of the algorithm is a set of samples that can beused to approximate the posterior distribution as follows:

${\hat{p}\left( x_{k} \middle| y_{1:k} \right)} = {\frac{1}{N}\;{\sum\limits_{i = 1}^{N}{{\delta\left( {x_{k} - x_{k}^{(i)}} \right)}.}}}$From these samples, any estimate of the system state can be calculated,such as the MMSE estimate,

${\hat{x}}_{k} = {{E\left\lbrack x_{k} \middle| y_{1:k} \right\rbrack} \approx {\frac{1}{N}{\sum\limits_{i = 1}^{N}{x_{k}^{(i)}.}}}}$

-   -   Similar expectations of the function g(x_(k)) (such as MAP        estimate, covariance, skewness, etc.) can be calculated as a        sample average.

General note: In the resampling stage, not only the particles but alsotheir respective SPKF propagated means and covariances are discarded orduplicated, i.e., we're resampling the whole parallel ensemble of SPKFs.The SPPF presented above makes use of a UKF for proposal generation. Ourpreferred form however, is a SPPF based around the square-root CDKF(SR-CDKF).

The UKF was used in the pseudo-code above in order to simplify thepresentation of the algorithm.

Algorithm 6: The Gaussian Mixture Sigma-Point Particle Filter (GMSPPF)

The full GMSPPF algorithm will now be presented based on the componentparts discussed above. As a graphical aid to understand this algorithm,please refer to the schematic presented in FIG. 12.

A) Time Update and Proposal Distribution Generation

At time k−1, assume the posterior state density is approximated by theG-component GMM

${{{\overset{\sim}{p}}_{G}\left( x_{k - 1} \middle| y_{1:{k - 1}} \right)} = {\sum\limits_{{\mathcal{g}} = 1}^{G}{{\overset{\sim}{\alpha}}_{k - 1}^{({\mathcal{g}})}\;{??}\mspace{11mu}\left( {{x_{k - 1};{\overset{\sim}{\mu}}_{k - 1}^{({\mathcal{g}})}},{\overset{\sim}{P}}_{k - 1}^{({\mathcal{g}})}} \right)}}},$and the process and observation noise densities are approximated by thefollowing I and J component GMMs respectively

$\begin{matrix}{{p_{??}\left( v_{k - 1} \right)} = {\sum\limits_{i = 1}^{I}{\beta_{k - 1}^{(i)}\;{??}\mspace{11mu}\left( {{v_{k - 1};\mu_{v,{k - 1}}^{(i)}},Q_{k - 1}^{(i)}} \right)}}} \\{{p_{??}\left( n_{k} \right)} = {\sum\limits_{j = 1}^{J}{\gamma_{k}^{(j)}\;{??}\mspace{11mu}\left( {{n_{k};\mu_{n,k}^{(j)}},R_{k}^{(j)}} \right)}}}\end{matrix}$

Following the GSF approach of [3], but replacing the EKF with a SPKF,the output of a bank of G″=GIJ parallel SPKFs are used to calculate GMMapproximations of p(x_(k)|y_(1:k−1)) and p(x_(k)|y_(1:k)) according tothe pseudo-code given below. For clarity of notation defineg′=g+(i−1)G,noting that references to g′ implies references to the respective g andi, since they are uniquely mapped. Similarly defineg″=g′+(j−1)GI,with the same implied unique index mapping from g″ to g′ and j. Thetime-update now proceeds as follows:

$\begin{matrix}1. & {{{{For}\mspace{14mu} j} = {1\mspace{11mu}\ldots\mspace{11mu} J}},{{{set}\mspace{14mu}{\overset{\sim}{p}\left( n_{k} \right)}^{(j)}} = {{{??}\left( {{n_{k};\mu_{n,k}^{(j)}},R_{k}^{(j)}} \right)}.}}} \\\; & {{{{For}\mspace{14mu} i} = {1\mspace{11mu}\ldots\mspace{11mu} I}},{{{set}\mspace{14mu}{\overset{\sim}{p}\left( v_{k - 1} \right)}^{(i)}} = {{??}\;\left( {{v_{k - 1};\mu_{v,{k - 1}}^{(i)}},Q_{k - 1}^{(i)}} \right)}}} \\\; & {{{{For}\mspace{14mu}{\mathcal{g}}} = {1\mspace{11mu}\ldots\mspace{11mu} G}},{{{set}\mspace{14mu}{\overset{\sim}{p}\left( x_{k - 1} \middle| y_{1:{k - 1}} \right)}^{({\mathcal{g}})}} = {{??}\;{\left( {{x_{k - 1};{\overset{\sim}{\mu}}_{k - 1}^{({\mathcal{g}})}},{\overset{\sim}{P}}_{k - 1}^{({\mathcal{g}})}} \right).}}}}\end{matrix}$

-   -   2. For g′=1 . . . G′ use the time update step of a SPKF        (employing the DSSM process equation x_(k)=f(x_(k−1), v_(k−1),        u_(k−1)) and densities {tilde over (p)}(x_(k−1)|y_(1:k−1))^((g))        and {tilde over (p)}(v_(k−1))^((i)) from above) to calculate a        Gaussian approximate predictive prior density

${{\overset{\sim}{p}\left( x_{k} \middle| y_{1:{k - 1}} \right)}^{({\mathcal{g}}^{\prime})} = {{??}\;\left( {{x_{k};\mu_{k|{k - 1}}^{({\mathcal{g}}^{\prime})}},P_{k|{k - 1}}^{({\mathcal{g}}^{\prime})}} \right)}},$and update the mixing weights:

$\alpha_{k|{k - 1}}^{({\mathcal{g}}^{\prime})} = {\frac{\alpha_{k - 1}^{({\mathcal{g}})}\beta_{k - 1}^{(i)}}{\sum\limits_{{\mathcal{g}} = 1}^{G}{\sum\limits_{i = 1}^{I}{\alpha_{k - 1}^{({\mathcal{g}})}\beta_{k - 1}^{(i)}}}}.}$

-   [3] ALSPACH, D. L., AND SORENSON, H. W. Nonlinear Bayesian    Estimation using Gaussian Sum Approximation. IEEE Transactions on    Automatic Control 17, 4 (1972), 439-448.    -   3. For g″=1 . . . . G″, complete the measurement update step of        each SPKF (employing the DSSM observation equation        y_(k)=h(x_(k), n_(k)), the current observation y_(k), and        densities {tilde over (p)}(x_(k)|y_(1:k−1))^((g′)) and        p(n_(k))^((j)) from above) to calculate a Gaussian approximate        posterior density        {tilde over (p)}(x _(k) |y _(1:k))^((g″)) =N(x _(k);μ_(k)        ^((g″)) ,P _(k) ^((g″))),        and update the GMM mixing weights:

${\alpha_{k}^{({\mathcal{g}}^{''})} = \frac{\alpha_{k|{k - 1}}^{({\mathcal{g}}^{\prime})}\gamma_{k}^{(j)}z_{k}^{(j)}}{\sum\limits_{{\mathcal{g}}^{\prime} = 1}^{G^{\prime}}{\sum\limits_{j = 1}^{J}{\alpha_{k}^{({\mathcal{g}}^{\prime})}\;\gamma_{k}^{(j)}\; z_{k}^{(j)}}}}},$where z_(k) ^((j))=p_(j)(y_(k)|x_(k)) is the observation likelihoodevaluated at x_(k)=μ_(k) ^((g′)) current observation, y_(k).

The predictive state density is now approximated by the following GMM:

${p_{??}\left( x_{k} \middle| y_{1:{k - 1}} \right)} = {\sum\limits_{{\mathcal{g}}^{\prime} = 1}^{G^{\prime}}{\alpha_{k|{k - 1}}^{({\mathcal{g}}^{\prime})}\;{??}\mspace{11mu}\left( {{x_{k};\mu_{k|{k - 1}}^{({\mathcal{g}}^{\prime})}},P_{k|{k - 1}}^{({\mathcal{g}}^{\prime})}} \right)}}$and the posterior state density (which will only be used as the proposaldistribution in the IS-based measurement update step) is approximated bythe following GMM:

${p_{??}\left( x_{k} \middle| y_{1:k} \right)} = {\sum\limits_{g^{''} = 1}^{G^{''}}{\alpha_{k}^{({\mathcal{g}}^{''})}\;{??}\mspace{11mu}{\left( {{x_{k};\mu_{k}^{({\mathcal{g}}^{''})}},P_{k}^{({\mathcal{g}}^{''})}} \right).}}}$

-   B) Measurement Update    -   1. Draw N samples {x_(k) ^((i); i=)1 . . . N} from the GMM        proposal distribution p_(G)(x_(k)|y_(1:k)) and calculate their        corresponding importance weights:

${\overset{\sim}{w}}_{k}^{(i)} = {\frac{{p\left( y_{k} \middle| x_{k}^{(i)} \right)}\;{p_{??}\left( x_{k}^{(i)} \middle| y_{1:{k - 1}} \right)}}{p_{??}\left( x_{k}^{(i)} \middle| y_{1:k} \right)}.}$

-   -   2. Normalize the weights:

$w_{k}^{(i)} = {\frac{{\overset{\sim}{w}}_{k}^{(i)}}{\sum\limits_{i = 1}^{N}{\overset{\sim}{w}}_{k}^{(i)}}.}$

-   -   3. Use one of the following approaches to fit a G-component GMM        to the set of weighted particles {w_(k) ^((i)), χ_(k) ^((i));        i=1 . . . N}, representing the updated GMM approximate state        posterior distribution at time k, i.e.

${{\overset{\sim}{p}}_{??}\left( x_{k} \middle| y_{1:k} \right)} = {\sum\limits_{{\mathcal{g}} = 1}^{G}{{\overset{\sim}{\alpha}}_{k}^{({\mathcal{g}})}\;{??}\mspace{11mu}{\left( {{x_{k};{\overset{\sim}{\mu}}_{k}^{({\mathcal{g}})}},{\overset{\sim}{P}}_{k}^{({\mathcal{g}})}} \right).}}}$

(Option A) First resample the set of weighted particles into a new setof N equally weighted particles using any of the efficient resamplingtechniques⁸ known and then apply an expectation-maximization (EM)algorithm to this new cloud of samples approximating the posteriordensity. ⁸Our preferred method is residual resampling [188, 46].

(Option B) Directly apply a weighted expectation-maximization (WEM)algorithm [132] to the weighted set of particles to recover the GMMposterior.

For both cases, the EM/WEM algorithm is “seeded” by the G means,covariances and mixing weights of the prior state GMM,p_(G)(x_(k−1)|y_(1:k−1)), and iterated until a certain convergencecriteria (such as relative dataset likelihood increase) is met.Convergence usually occur within 4-6 iterations. Alternatively, asdiscussed earlier, an adaptive model-order-selection EM/WEM approach canbe used to adaptively determine the optimal number of Gaussian componentdensities needed in the posterior GMM to accurately model the posteriorcloud of samples.

-   C) Inference

Once the full posterior state density has been calculated in theprevious step, any “optimal” estimate of the underlying system state canbe calculated. These include estimates such as the condition mean(MMSE), maximum a-posteriori (MAP), mode, median, etc. to name but afew. As an example, we present the MMSE estimate:

-   [132] MCLACHLAN, G., AND KRISHNAN, T. The EM Algorithm and    Extensions. Wiley, 1997.

The conditional mean state estimate{circumflex over (x)} _(k) =E[x _(k) |y _(1:k)],and the corresponding error covariance{circumflex over (P)} _(k) =E[(x _(k) −{circumflex over (x)} _(k))(x_(k) −{circumflex over (x)} _(k))^(T)],can be calculated in one of two ways:

(Option A) The estimates can be calculated before the EM/WEM smoothingstage by a direct weighted sum of the particle set,

$\begin{matrix}{\hat{x} = {\sum\limits_{i = 1}^{N}{w_{k}^{(i)}\; x_{k}^{(i)}}}} \\{{{\hat{P}}_{k} = {\sum\limits_{i = 1}^{N}{w_{k}^{(i)}\;\left( {x_{k}^{(i)} - {\hat{x}}_{k}} \right)\;\left( {x_{k}^{(i)} - {\hat{x}}_{k}} \right)^{T}}}},}\end{matrix}$

(Option B) or after the posterior GMM has been fitted,

$\begin{matrix}{{\hat{x}}_{k} = {\sum\limits_{g = 1}^{G}{{\overset{\sim}{\alpha}}_{k}^{(g)}\mspace{11mu}{\overset{\sim}{\mu}}_{k}^{(g)}}}} \\{{\hat{P}}_{k} = {\sum\limits_{g = 1}^{G}{{{\overset{\sim}{\alpha}}_{k}^{(g)}\;\left\lbrack {P_{k}^{(g)} + {\left( {{\overset{\sim}{\mu}}_{k}^{(g)} - {\hat{x}}_{k}} \right)\mspace{11mu}\left( {{\overset{\sim}{\mu}}_{k}^{(g)} - {\hat{x}}_{k}} \right)^{T}}} \right\rbrack}.}}}\end{matrix}$

Since N>>G, the first approach (Option A) is computationally moreexpensive than the second, but possibly generates better (lowervariance) estimates, since it calculates the estimates before theimplicit resampling of the WEM step. The choice of which method to usewill depend on the specifics of the inference problem and is an ongoingresearch question.

It will be obvious to those having skill in the art that many changesmay be made to the details of the above-described embodiments withoutdeparting from the underlying principles of the invention. The followingare several examples of such changes.

Although the preferred embodiment estimated a single time-varying IMUgyroscope rate bias vector, the invention can also be applied toestimate IMU scale and alignment parameters. Additional parameters areaugmented into the state vector.

Although the preferred embodiment neglected the effects resulting fromthe rotational rate of the earth as measured in the navigation frame(Coriolis effect), the invention can also be applied to account forthese effects by appropriate modification of the kinematics equations.

Although the preferred embodiment used a quaternion representation ofthe attitude, the invention can also be applied to other attituderepresentations such as standard Euler angles.

Although the preferred embodiment used a Lagrange approach to maintainunity norm on the quaternion, the constraint may also be accomplished byestimating and updating an error quaternion.

The scope of the present invention should, therefore, be determined onlyby the following claims.

1. A method performed in an integrated navigation system to estimate thenavigational state of an object, the navigational state characterized byrandom variables of a kinematics-based system state space model, thesystem state space model specifying a time evolution of the system andits relationship to sensor observations, comprising: acquiringobservation data produced by measurement sensors that provide noisyinformation about the navigational state, the measurement sensorsincluding an inertial measurement unit (IMU) and a global positioningsystem (GPS) operating to provide information for estimating a set ofnavigational state components that include position, velocity, attitude,and angular velocity; and providing a probabilistic inference system tocombine the observation data with prediction values of the system statespace model to estimate the navigational state, the probabilisticinference system implemented to include a realization of a Gaussianapproximate random variable propagation technique performingdeterministic sampling without analytic derivative calculations.
 2. Themethod of claim 1, in which the navigational state component setincludes bias and scale factor information associated with the IMU. 3.The method of claim 1, in which the navigational state component setincludes phase and timing error information associated with the GPS. 4.The method of claim 1, in which the measurement sensors producingobservation data constitute sensors of multiple orbital GPS satellitesfrom which raw satellite signal transmissions propagate.
 5. The methodof claim 1, in which the random variable propagation technique of theprobabilistic inference system used to estimate the navigational stateincludes: using mean values and square root decomposition of acovariance matrix of a prior random variable to deterministicallycalculate a set of sigma-points and an associated set of weight values;and propagating the calculated set of weighted sigma-points through thesystem state space model to produce a posterior set of weightedsigma-points used to calculate posterior statistics.
 6. The method ofclaim 5, in which the calculation of posterior statistics entails theuse of closed form functions of the propagated sigma-points and weightvalues.
 7. The method of claim 5, further comprising using thecalculated posterior statistics to give an approximately optimal updateof the navigational state and its covariance by combining the predictionvalues of the system state space model and the sensor observations. 8.The method of claim 5, in which the probabilistic inference system usedto estimate the navigational state is implemented with one of unscentedKalman filter (UKF) algorithms, central difference Kalman filter (CDKF)algorithms, square root version of UKF, square root version of CDKF,sigma-point particle filter (SPPF) algorithms, or Gaussian mixturesigma-point particle filter (GMSPPF) algorithms.
 9. The method of claim1, in which the observation data provide noisy information about thenavigational state at a time that lags a current time because of sensorlatency, communication delay, or other processing delay, and furthercomprising: maintaining a cross correlation between a navigational stateat a current time and the navigational state to which the latency laggedobservation data correspond; and using the cross correlation tooptimally combine latency lagged observation data with the predictionvalues of the system state space model.
 10. The method of claim 1, inwhich the GPS operates to receive raw satellite signal transmissions,and in which the received raw satellite signal transmissions areintegrated directly with the IMU, thereby to form a tightly coupledintegrated navigation system.
 11. The method of claim 1, in which theposition and velocity information provided by the GPS is integrated withthe IMU, thereby to form a loosely coupled integrated navigation system.12. An integrated navigation system for estimating the navigationalstate of an object, the navigational state characterized by randomvariables of a kinematics-based system state space model, the s stemstate space model specifying a time evolution of the system and itsrelationship to sensor observations, comprising: measurement sensorsproducing observation data in the form of noisy information about thenavigational state, the measurement sensors including an integratedmeasurement unit (IMU) and a global positioning system (GPS) operatingto provide information for estimating a set of navigational statecomponents that include position, velocity, attitude, and angularvelocity; and a probabilistic inference system configured to combine theobservation data with prediction values of the system state space modelto estimate the navigational state, the probabilistic inference systemimplemented to include a realization of a Gaussian approximate randomvariable propagation technique performing deterministic sampling withoutanalytic derivative calculations.
 13. The system of claim 12, in whichthe observation data provide noisy information about the navigationalstate at a time that lags a current time because of sensor latency,communication delay, or other processing delay, and further comprising:a computational subsystem operable to maintain a cross correlationbetween a navigational state at a current time and the navigationalstate to which the latency lagged observation data correspond; and acomputational subsystem operable to use the cross correlation tooptimally combine latency lagged observation data with the predictionvalues of the system state space model.
 14. The system of claim 12, inwhich the navigational state component set includes bias and scalefactor information associated with the IMU.
 15. The system of claim 12,in which the navigational state component set includes phase and timingerror information associated with the GPS.
 16. The system of claim 12,in which the measurement sensors producing observation data constitutesensors of multiple orbital GPS satellites from which raw satellitesignal transmissions propagate.
 17. The system of claim 12, in which therandom variable propagation technique of the probabilistic inferencesystem used to estimate the navigational state includes: use of meanvalues and square root decomposition of a covariance matrix of a priorrandom variable to deterministically calculate a set of sigma-points andan associated set of weight values; and propagation of the calculatedset of weighted sigma-points through the system state space model toproduce a posterior set of weighted sigma-points used to calculateposterior statistics.
 18. The system of claim 17, in which thecalculation of posterior statistics entails the use of closed formfunctions of the propagated sigma-points and weight values.
 19. Thesystem of claim 17, in which the random variable propagation techniquefurther comprises use of the calculated posterior statistics to give anapproximately optimal update of the navigational state and itscovariance by combining the prediction values of the system state spacemodel and the sensor observations.
 20. The system of claim 17, in whichthe probabilistic inference system used to estimate the navigationalstate is implemented with one of unscented Kalman filter (UKF)algorithms, central difference Kalman filter (CDKF) algorithms, squareroot version of UKF, square root version of CDKF, sigma-point particlefilter (SPPF) algorithms, or Gaussian mixture sigma-point particlefilter (GMSPPF) algorithms.
 21. The system of claim 12, in which the GPSoperates to receive raw satellite signal transmissions, and in which thereceived raw satellite signal transmissions are integrated directly withthe IMU, thereby to form a tightly coupled integrated navigation system.22. The system of claim 12, in which the position and velocityinformation provided by the GPS is integrated with the IMU, thereby toform a loosely coupled integrated navigation system.
 23. A methodperformed in a global positioning system (GPS) to estimate thenavigational state of an object, the navigational state characterized byrandom variables of a kinematics-based system state space model, thesystem state space model specifying a time evolution of the system andits relationship to sensor observations, comprising: acquiringobservation data produced by measurement sensors that provide noisyinformation about the navigational state, the measurement sensorsincluding a GPS operating to receive raw satellite signal transmissionsand provide information for estimating a set of navigational statecomponents including position and velocity; and providing aprobabilistic inference system to combine the observation data withprediction values of the system state space model to estimate thenavigational state, the probabilistic inference system implemented toinclude a realization of a Gaussian approximate random variablepropagation technique performing deterministic sampling without analyticderivative calculations.
 24. The method of claim 23, in which thenavigational state component set includes phase and timing errorinformation associated with the GPS.
 25. The method of claim 23, inwhich the measurement sensors producing observation data constitutesensors of multiple orbital GPS satellites from which raw satellitesignal transmissions propagate.
 26. The method of claim 23, in which therandom variable propagation technique of the probabilistic inferencesystem used to estimate the navigational state includes: using meanvalues and square root decomposition of a covariance matrix of a priorrandom variable to deterministically calculate a set of sigma-points andan associated set of weight values; and propagating the calculated setof weighted sigma-points through the system state space model to producea posterior set of weighted sigma-points used to calculate posteriorstatistics.
 27. The method of claim 26, in which the calculation ofposterior statistics entails the use of closed form functions of thepropagated sigma-points and weight values.
 28. The method of claim 26,further comprising using the calculated posterior statistics to give anapproximately optimal update of the navigational state and itscovariance by combining the prediction values of the system state spacemodel and the sensor observations.
 29. The method of claim 26, in whichthe probabilistic inference system used to estimate the navigationalstate is implemented with one of unscented Kalman filter (UKF)algorithms, central difference Kalman filter (CDKF) algorithms, squareroot version of UKF, square root version of CDKF, sigma-point particlefilter (SPPF) algorithms, or Gaussian mixture sigma-point particlefilter (GMSPPF) algorithms.
 30. A method of estimating a state of asystem, the state characterized by random variables of a system statespace model, the system state space model specifying a time evolution ofthe system and its relationship to sensor observations, comprising:acquiring observation data produced by measurement sensors that providenoisy information about the state of the system at a time that lags acurrent time because of sensor latency, communication delay, or otherprocessing delay; providing a probabilistic inference system to combinethe observation data with prediction values of the system state spacemodel to estimate the state of the system, the probabilistic inferencesystem implemented to include a realization of a Gaussian approximaterandom variable propagation technique performing deterministic samplingwithout analytic derivative calculations; maintaining a crosscorrelation between a state of the system at a current time and thestate to which the latency lagged observation data correspond; and usingthe cross correlation to optimally combine latency lagged observationdata with the prediction values of the system state space model.
 31. Themethod of claim 30, in which the state of a system is a navigationalstate.